DOCSLIB.ORG

Polymers Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Polymers Physics Polymers Physics Minne Paul Lettinga Forschungszentrum J¨ulich Institute for Complex Systems - 3 J¨ulich, July 2011 2 Contents 1 Ideal polymer chain 3 1.1 Definition of an ideal polymer: . .3 1.2 End-to-end vector . .3 1.3 Radius of Gyration . .5 1.4 Measuring polymer size . .6 1.5 Probability distribution . .6 1.6 Entropic spring . .8 2 Non-ideal polymers 11 2.1 Fixed valency angle polymers . 11 2.2 Persistence length of a polymer chain . 13 2.3 Kuhn segments . 15 2.4 Occupied volume . 15 2.5 Obtaining persistence length experimentally . 17 2.6 Gaussian Chain Model . 19 2.6.1 Appendix A . 21 3 Polymer Dynamics 23 3.1 Rouse model . 23 Literature i 1 2 Chapter 1 Ideal polymer chain 1.1 Definition of an ideal polymer: • no volume per joint • no interactions • Freely jointed chain The consequences of these assumptions are manyfold. For example will be no interaction also between polymer coils, so that for example with ideal polymers no rubber can be formed, see later. We start with ideal polymers because it is easy to derive a few basic features of polymers. 1.2 End-to-end vector The first feature we will calculate is the average end-to-end distance R~ between the start and ~ ~ the end of the polymer, which consists of N segments b with orientation ~ui and length b. R is a measure of the size of the polymer, but not something one could readily measure. Because R~ can point in any direction, when R~ is averaged over all these directions it will give hR~i = 0, which is the same issue as we had with the displacement of colloidal particles. As with colloidal particles the first non-zero parameter is N N N N 2 X X X X R = ( ~ui)( ~uj) = ~ui~uj; (1.1) i=1 j=1 i=1 j=1 so the average of this quantity gives 3 R j Rij rj b i a i b ri Figure 1.1: (a) Sketch of a freely jointed chain polymer. R~ shows the end-to-end vector of this polymer. (b) Illustration of a subfragment (in red) for which the end-to-end distance needs to be calculated. 4 N N N N N ~ 2 X X X 2 X X 2 hR i ≡ h~ui~uji = h~ui i + h~ui~uji = Nb = Lb; (1.2) i=1 j=1 i=1 i=1 j=1;6=i PN PN where L = Nb is the contour length of the polymer. We used here that i=1 j=1;6=ih~ui~uji = 0 because the angles of segments i and j are not correlated. The end to vector is thus given by R = phR2i = N 1=2b. From this we see that ideal chains form entangled coils. 1.3 Radius of Gyration The radius of gyration Rg is a quantity that can be more readily measured, as we will see later on. Rg is defined as the average distance to the center of mass of the object, in this case the polymer coil. As with the end-to-end vector R a straightforward average of the vector ~ PN Rg = i=1(~ri − ~rcm) would give zero, so again we have to calculate the square of the distance 1 PN of segment i with position ~r to the center of mass ~rcm = N j=1 ~rj. N N N N N 1 X 1 X X 1 X X R2 = h (~r − ~r )2i = h(~r − ~r )2i == hR~ 2 i: (1.3) g N i cm 2N 2 i j 2N 2 ij i=1 i=1 j=1 i=1 j=1 Since vecRij = ~ri − ~rj is again an end-to-end distance but now from the full polymer but from a subfragment of the polymer, see Fig. 1.1b, we can use the result EQ. 1.2 to calculate vecRij. Here we assume that the number of segments subfragment is large, i.e. ji − jj 1, which will be the case for big polymers: 2 2 h(~ri − ~rj) i = ji − jjb : (1.4) Therefore N N b2 X X R2 = ji − jj: (1.5) g 2N 2 i=1 j=1 For large N this sum can be replaced by an integral. We leave it as an exercise to show now that 1 R2 = Nb2: (1.6) g 6 So the ratio of the radius of gyration and the average end-to-end vector is 1=6 for ideal chains. 5 1.4 Measuring polymer size In the past sections we have always considered average distances between the different segments of the polymer. Suppose now that light can be scattered from each node between the segments. This would then be very similar to scattering from an ensemble of point sources as has been treated for colloids earlier. There it was shown how the structure factor of the ensemble can be obtained from light scattering: N N 1 X X S(~k) = hexp(i~k · (~r − ~r ))i; (1.7) N i j i=1 j=1 ~ ~ where k is the scattering vector with jkj = 2π/λ and ~ri are the positions of the colloids. We can now use exactly the same expression to calculated the intra-polymer structure factor taking now for ~ri the locations of the nodes, wich is also called the form factor. Doing so we immediately recognize ~ 2 2 hexp(ik · (~ri − ~rj))i = exp(−k ji − jjb =6) (1.8) This gives Z N Z N ~ 1 2 2 S(k) = di dj exp(−k ji − jjb =6 = Nf(kRg) (1.9) N 0 0 where f(x) = 2(exp −x2) − 1 + x2)=x4. For small wave vectors it can be shown that ~ ~ 2 2 S(k) = S(k)(1 − k Rg=3) (1.10) 1.5 Probability distribution The probability P (R;N) to find a chain with end-to-end vector R for a chain with N segments is given by a summation of the z possible position the chain can have before the last step from N − q to N. Of course each of these positions have there own probability P (R − bi;N − 1): z 1 X P (R;N) = P (R − b ;N − 1) (1.11) z i i=1 6 Since the chains are very long , i.e. N 1 and jRj jbij we can use the Taylor expansion to two variables, one of which is a vector: @f(r; a) f(r + dr; a + da) = f(r) + dr · f(r; a) + da (1.12) Or @a 1 1 @2f(r; a) + drdr : f(r; a) + da2 + O(3) 2 OrOr 2 @a2 We use this expansion to find an expression for P (R − bi;N − 1) in terms of P (R;N). In the case of the ideal chain da = −1 and dr = −bi so that we have now z 1 X @P (R;N) P (R;N) = fP (R;N) − b · P (R;N) − (1.13) z i Or @N i=1 1 1 @2P (R;N) + b b : P (R;N) + :::g 2 i i OrOr 2 @N 2 Let us look at each of the terms. First note that all the terms independent of i can be taken out of the summation, i.e. the first, third and fifth term. The last term can be neglected, assuming that N 1. Next, note that the second term in the equation disappears because the last step i towards the end of the polymer can come from every direction, so z X bi = 0: (1.14) i=1 1 Pz The fourth term is less straightforward to deal with. Apparently for z i=1 bibi we have z z 3 1 X 1 X X @ @ bibi : OrOr = bi,αbi,β z z @rα @rβ i=1 i=1 α,β=1 3 2 2 2 X @ @ b @ = b δαβ = 2 (1.15) @rα @rβ 3 @R α,β=1 In making the second step in the derivation we assumed that the polymer is located on a rectangular lattice. Using Eq. 1.14 and 1.15 in Eq. 1.13 results in the differential equation @P (R;N) b2 @2P (R;N) = ; (1.16) @N 6 @R2 7 Which has the solution, 3 −3R2 P (R;N) = ( )3=2 exp 2πNb2 2Nb2 3 −3R2 = ( )3=2 exp (1.17) 2πhR2i 2hR2i As you see, this is a Gaussian distribution of the probabilities. It is important to note that any model can be used to calculate hR2i, which we will use later when we discuss non-ideal polymers. 1.6 Entropic spring The probability density to find a polymer coil with an end-to-end vector length R can be directly related to the entropy of the coil, which is given by the number of ways WN (R) to obtain a coil with an end-to-end vector R: S(R) = k ln WN (R); (1.18) where k is the Boltzmann factor. WN (R) is directly related to the probability density P (R;N) to find a polymer with N monomers with a size R via some constant, so that S(R) = k ln P (R;N) + const: (1.19) 3 = − kR2=Lb + const: (1.20) 2 where we used the general result for a Gaussian chain, Eq. 1.17. Knowing the entropy of the chain, the free energy F = E − TS can be readily calculated. Since we are dealing with an ideal polymer, there are no interactions between the polymer segments so that E = 0 and F = ln P (R;N) + const: (1.21) 3kT R2 = + const: (1.22) 2Lb Now suppose that the chain is submitted to a force f~, resulting in small changedR~ of R.
Load more

Recommended publications
  • Ideal Chain Conformations and Statistics
    ENAS 606 : Polymer Physics Chinedum Osuji 01.24.2013:HO2 Ideal Chain Conformations and Statistics 1 Overview Consideration of the structure of macromolecules starts with a look at the details of chain level chemical details which can impact the conformations adopted by the polymer. In the case of saturated carbon ◦ backbones, while maintaining the desired Ci−1 − Ci − Ci+1 bond angle of 112 , the placement of the final carbon in the triad above can occur at any point along the circumference of a circle, defining a torsion angle '. We can readily recognize the energetic differences as a function this angle, U(') such that there are 3 minima - a deep minimum corresponding the the trans state, for which ' = 0 and energetically equivalent gauche− and gauche+ states at ' = ±120 degrees, as shown in Fig 1. Figure 1: Trans, gauche- and gauche+ configurations, and their energetic sates 1.1 Static Flexibility The static flexibility of the chain in equilibrium is determined by the difference between the levels of the energy minima corresponding the gauche and trans states, ∆. If ∆ < kT , the g+, g− and t states occur with similar probability, and so the chain can change direction and appears as a random coil. If ∆ takes on a larger value, then the t conformations will be enriched, so the chain will be rigid locally, but on larger length scales, the eventual occurrence of g+ and g− conformations imparts a random conformation. Overall, if we ignore details on some length scale smaller than lp, the persistence length, the polymer appears as a continuous flexible chain where 1 lp = l0 exp(∆/kT ) (1) where l0 is something like a monomer length.
    [Show full text]
  • Properties of Two-Dimensional Polymers
    Macromolecules 1982,15, 549-553 549 The squared frequencies uA2are obtained from the ei- (4) Flory, P. J. "Statistical Mechanics of Chain Molecules"; In- genvalues Of this matrix described in the text* terscience: New York, 1969; p 315. (5) Volkenstein,M. V. &Configurational Statistics of polymeric for N = 3 and 4 will be found in ref 7. Chains" (translated from the Russian edition by S. N. Ti- masheff and M. J. Timasheff);Interscience: New York, 1963; References and Notes p 450. (1) Weiner, J. H.; Perchak, D. Macromolecules 1981, 14, 1590. (6) Frenkel, J. "Kinetic Theory of Liquids"; Oxford University (2) Weiner, J. H., Macromolecules, preceding paper in this issue. Press: London, 1946, Dover reprint, 1955; pp 474-476. (3) G6, N.; Scheraga, H. A. Macromolecules 1976, 9, 535. (7) Perchak, D. Doctoral Dissertation, Brown University, 1981. Properties of Two-Dimensional Polymers Jan Tobochnik and Itzhak Webman* Department of Mathematics, Rutgers, The State University, New Brunswick, New Jersey 08903 Joel L. Lebowitzt Institute for Advanced Study, Princeton, New Jersey 08540 M. H. Kalos Courant Institute of Mathematical Sciences, New York university, New York, New York 10012. Received August 3, 1981 ABSTRACT: We have performed a series of Monte Carlo simulations for a two-dimensional polymer chain with monomers interacting via a Lennard-Jones potential. An analysis of a mean field theory, based on approximating the free energy as the sum of an elastic part and a fluid part, shows that in two dimensions there is a sharp collapse transition, at T = 0, but that there is no ideal or quasi-ideal behavior at the transition as there is in three dimensions.
    [Show full text]
  • Development of the Semi-Empirical Equation of State for Square-Well Chain Fluid Based on the Statistical Associating Fluid Theory (SAFT)
    Korean J. Chem. Eng., 17(1), 52-57 (2000) Development of the Semi-empirical Equation of State for Square-well Chain Fluid Based on the Statistical Associating Fluid Theory (SAFT) Min Sun Yeom, Jaeeon Chang and Hwayong Kim† School of Chemical Engineering, Seoul National University, Shinlim-dong, Kwanak-ku, Seoul 151-742, Korea (Received 25 August 1999 • accepted 29 October 1999) Abstract−A semi-empirical equation of state for the freely jointed square-well chain fluid is developed. This equation of state is based on Wertheim's thermodynamic perturbation theory (TPT) and the statistical associating fluid theory (SAFT). The compressibility factor and radial distribution function of square-well monomer are ob- tained from Monte Carlo simulations. These results are correlated using density expansion. In developing the equa- tion of state the exact analytical expressions are adopted for the second and third virial coefficients for the com- pressibility factor and the first two terms of the radial distribution function, while the higher order coefficients are determined from regression using the simulation data. In the limit of infinite temperature, the present equation of state and the expression for the radial distribution function are represented by the Carnahan-Starling equation of state. This semi-empirical equation of state gives at least comparable accuracy with other empirical equation of state for the square-well monomer fluid. With the new SAFT equation of state from the accurate expressions for the mo- nomer reference and covalent terms, we compare the prediction of the equation of state to the simulation results for the compressibility factor and radial distribution function of the square-well monomer and chain fluids.
    [Show full text]
  • High Elasticity of Polymer Networks. Polymer Networks
    High Elasticity of Polymer Networks. Polymer Networks. Polymer network consists of long polymer chains which are crosslinked with each other to form a giant three-dimensional macromolecule. All polymer networks except those who are in the glassy or partially crystalline states exhibit the property of high elasticity, i.e., the ability to undergo large reversible deformations at relatively small applied stress. High elasticity is the most specific property of polymer materials, it is connected with the most fundamental features of ideal chains considered in the previous lecture. In everyday life, highly elastic polymer materials are called rubbers. Molecular Nature of the High Elasticity. Elastic response of the crystalline solids is due to the change of the equilibrium interatomic distances under stress and therefore, the change in the internal energy of the crystal. Elasticity of rubbers is composed from the elastic responses of the chains crosslinked in the network sample. External stress changes the equilibrium end-to-end distance of a chain, and it thus adopts a less probable conformation. Therefore, the elasticity of rubbers is of purely entropic nature. Typical Stress-Strain Curves. for steel for rubbers А - the upper limit for stress-strain linearity, В - the upper limit for the reversibility of deformations, С - the fracture point. - the typical values of deformations ∆ l l are much larger for rubber; - the typical values of strain σ are much larger for steel; - the typical values of the Young modulus is enormously larger for steel than for rubber ( E 1 0 1 1 Pa and 1 0 5 − 1 0 6 Pa, respectively); - for steel linearity and reversibility are lost practically simultaneously, while for rubbers there is a very wide region of nonlinear reversible deformations; - for steel there is a wide region of plastic deformations (between points B and C) which is all but absent for rubbers.
    [Show full text]
  • Introduction to Polymer Physics Lecture 1 Boulder Summer School July – August 3, 2012
    Introduction to polymer physics Lecture 1 Boulder Summer School July – August 3, 2012 A.Grosberg New York University Lecture 1: Ideal chains • Course outline: • This lecture outline: •Ideal chains (Grosberg) • Polymers and their •Real chains (Rubinstein) uses • Solutions (Rubinstein) •Scales •Methods (Grosberg) • Architecture • Closely connected: •Polymer size and interactions (Pincus), polyelectrolytes fractality (Rubinstein) , networks • Entropic elasticity (Rabin) , biopolymers •Elasticity at high (Grosberg), semiflexible forces polymers (MacKintosh) • Limits of ideal chain Polymer molecule is a chain: Examples: polyethylene (a), polysterene •Polymericfrom Greek poly- (b), polyvinyl chloride (c)… merēs having many parts; First Known Use: 1866 (Merriam-Webster); • Polymer molecule consists of many elementary units, called monomers; • Monomers – structural units connected by covalent bonds to form polymer; • N number of monomers in a …and DNA polymer, degree of polymerization; •M=N*mmonomer molecular mass. Another view: Scales: •kBT=4.1 pN*nm at • Monomer size b~Å; room temperature • Monomer mass m – (240C) from 14 to ca 1000; • Breaking covalent • Polymerization bond: ~10000 K; degree N~10 to 109; bondsbonds areare NOTNOT inin • Contour length equilibrium.equilibrium. L~10 nm to 1 m. • “Bending” and non- covalent bonds compete with kBT Polymers in materials science (e.g., alkane hydrocarbons –(CH2)-) # C 1-5 6-15 16-25 20-50 1000 or atoms more @ 25oC Gas Low Very Soft solid Tough and 1 viscosity viscous solid atm liquid liquid Uses Gaseous
    [Show full text]
  • Chemistry C3102-2005: Polymers Section
    Chemistry C3102-2005: Polymers Section Dr. Edie Sevick, Research School of Chemistry, ANU 2.0 Thermodynamics of an Ideal Chain: Stretching and Squashing In this section, we begin to investigate the thermodynamics of polymers, starting first, with an isolated ideal chain. We begin with the general relation ∆F = ∆U T ∆S (1) − where the first term on the RHS is the change in the free energy, ∆U is the change in the internal energy of the chain, and ∆S is the change in entropy. Examples of interactions which contribute to ∆U range from Lennard-Jones interactions to complex interactions which scientists might arbitrarily propose, all of which are of the form U = f(rij) where rij is a vector of distances between monomers, solvent, or other particle or surface which might be included in the system. By definition, an ideal chain suffers NO interactions and contributes nothing to the internal energy. Consequently, if we take an ideal chain and end-tether it to a phantom wall, or change the solvent, then the change in the internal energy of the system is 0. On the other hand, the chain has considerable disorder, measured by the number of configurations that it can adopt and quantified by the thermodynamic quantity entropy. If we were to hold the two ends of the chain a distance Na apart so that the chain was taut, we would considerably decrease the number of configurations that the chain could adopt. If S1 is the entropy of the ideal chain in its relaxed, or natural state, and S2 the entropy of the chain in its taut state, then S1 >> S2 or ∆S = S2 S1 would be negative, and by the above equation, ∆F , would be positive.
    [Show full text]
  • Chapter #2 Thermodynamics of an Ideal Chain
    Chemistry C3102-2006: Polymers Section Dr. Edie Sevick, Research School of Chemistry, ANU 2.0 Thermodynamics of an Ideal Chain: Stretching and Squashing In this section, we begin to investigate the thermodynamics of polymers, starting first, with an isolated ideal chain. We begin with the Helmholtz free energy, F , which corresponds to the available work at a constant temperature: ∆F = ∆U T ∆S, (1) − where U is the internal energy of the chain, T is the temperature which is held constant and S is the entropy. Examples of energies which contribute to U range from Lennard-Jones interactions to complex interactions which scientists might propose, all of which are of the form U = f(rij) where rij is a vector of distances between monomers, solvent, or other particle or surface which might be included in the system. By definition, an ideal chain suffers NO interactions and contributes nothing to the internal energy at a constant temperature. Consequently, if we take an ideal chain and end-tether it to a phantom wall, or change the solvent, then the change in the internal energy of the system is 0 under isothermal conditions: ∆U = 0 1. On the other hand, the chain has considerable disorder, measured by the number of configurations that it can adopt and quantified by the thermodynamic quantity entropy. If we were to hold the two ends of the chain a distance Na apart so that the chain was taut, we would considerably decrease the number of configurations that the chain could adopt. If S1 is the entropy of the ideal chain in its relaxed, or natural state, and S2 the entropy of the chain in its taut state, then S1 >> S2 or ∆S = S2 S1 would be negative, and by the above equation, ∆F , would be − positive.
    [Show full text]
  • Lecture 8: Polymer Solutions – Flory-Huggins Theory
    Prof. Tibbitt Lecture 8 Networks & Gels Lecture 8: Polymer solutions { Flory-Huggins Theory Prof. Mark W. Tibbitt { ETH Z¨urich { 19 M¨arz2019 1 Suggested reading • Molecular Driving Forces { Dill and Bromberg: Chapter 32 • Polymer Physics { Rubinstein and Colby: Chapters 4,5 2 Flory-Huggins Theory In the last lecture, we developed the regular solution theory from a lattice model combining the entropy and energy of mixing to calculate the free energy of mixing for regular solutions of two species with equal molecular volume. We now want to develop a theory that can account for the fact that we do not always consider only molecules that are of equivalent volume and equal volume to the lattice site. The includes the mixing of polymer chains and here we want to consider how our previous calculations change when we consider the free energy of mixing for polymers. Ideally, we would like a general approach that can handle binary polymer-polymer solutions as well as polymer-solvent solutions. To do this we will follow the method developed by Flory and Huggins known as the Flory-Huggins Theory and follow a lattice approach with a mean-field estimate as we did for the regular solution theory. We will calculate the entropy of mixing and the energy of mixing and combine these terms to develop a formulation for the free energy of mixing. Here we define the system similarly as a lattice consisting of N sites of equal volume v0. We take n1 and n2 as the number of molecules of species 1 and 2, respectively, with degree of polymerizations x1 and x2, monomer volumes v1 and v2 and total volumes V1 and V2.
    [Show full text]
  • Solution Exercise 11 HS 2015
    Statistical Thermodynamics Solution Exercise 11 HS 2015 Solution Exercise 11 Problem 1: Ideal chain models a) The end-to-end vector of a polymer chain is defined as n ~ X Rn = ~ri : (1.1) i=1 Since there are no correlations between the individual ~ri all chain orientations will be equally likely. The mean end-to-end vector is therefore equal to zero. ~ hRni = 0 : (1.2) the mean-square end-to-end distance, however, provides a more useful description of the system. With *0 n 1 0 n 1+ 2 ~ 2 ~ ~ X X hR i = hRni = hRn · Rni = @ ~riA · @ ~rjA ; (1.3) i=1 j=1 and with the hint given one gets n n 2 2 X X hR i = l hcos θiji : (1.4) i=1 j=1 Again we use the assumption that there are no correlations between the orientations of the bond vectors. For all i 6= j this will lead to hcos θiji = 0, however for i = j (one and the same bond) θii will always be equal to 0° which leads to hcos θiii = 1. With this Eq. (1.4) then simplifies to hR2i = nl2 : (1.5) b) When considering correlations between the individual bond vectors the hcos θiji are not necessarily equal to zero or one anymore. In general one might assume, however, that for bonds that are infinitely far apart from each other there won't be any correlations, i.e lim hcos θiji = 0 : (1.6) ji−jj!1 Without any further knowledge about the polymer we cannot write an analytic expression for the mean-square end-to-end distance apart from the most general form in which the correlations between the ith element and all the other elements in the chains is denoted as n 0 X Ci ≡ hcos θiji : (1.7) j=1 With these considerations Eq.
    [Show full text]
  • IV. Polymer Solutions and Blends
    IV. Polymer Solutions and Blends Thermodynamics of mixture of n1 and n2 moles of each component is usually analyzed in terms of Gibbs free energy G. Change in the free energy: dG Vdp SdT idni i1,2 Here V is the volume, p is the pressure, S is the entropy and i is the chemical potential of component i. Entropy of mixing: The total number of configurations for m1 and m2 number of molecules on a lattice with each site occupied (total number of sites m) m! Sm k ln k ln km1 ln x1 m2 ln x2 m1!m2! Sm Sm S1 S2 km1 ln x1 m2 ln x2 Rx1 ln x1 x2 ln x2 Here xi is the molar fraction of the component i. Because 0< xi<1, Sm >0 and configurational entropy always favor mixing. 1 Enthalpy of mixing: for pure components Hi=mizwii/2, z is number of neighbors, wii is the interaction energy between two molecules (negative). Assuming no change in V or p with mixing (lattice approach), and completely random mixing: 1 1 H m z(x w x w ) m z(x w x w ) m 2 1 1 11 2 12 2 2 1 12 2 22 m1m2 H m H m H1 H 2 zw; where w=w -(w +w )/2 m 12 11 22 zw To characterize enthalpy of mixture the interaction parameter is introduced: kT H m x1x2 kT ; per site (or per molecule) and H m x 1 x 2 RT ; per mole. Thus the free energy of mixing per site: Gm (x1 ln x1 x2 ln x2 x1x2 )kT Flory-Huggins Theory Is extension of regular solution theory to the case of polymers.
    [Show full text]
  • Polymer Solutions: an Introduction to Physical Properties
    Polymer Solutions: An Introduction to Physical Properties. Iwao Teraoka Copyright © 2002 John Wiley & Sons, Inc. ISBNs: 0-471-38929-3 (Hardback); 0-471-22451-0 (Electronic) POLYMER SOLUTIONS POLYMER SOLUTIONS An Introduction to Physical Properties IWAO TERAOKA Polytechnic University Brooklyn, New York A JOHN WILEY & SONS, INC., PUBLICATION Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc., is aware of a claim, the product names appear in initial capital or ALL CAPITAL LETTERS. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Copyright © 2002 by John Wiley & Sons, Inc., New York. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. ISBN 0-471-22451-0 This title is also available in print as ISBN 0-471-38929-3.
    [Show full text]
  • Quantum Mechanics Polymer Physics
    Quantum Mechanics_Polymer physics Polymer physics is the field of physics that studies polymers, their fluctuations,mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively.[1][2][3][4] While it focuses on the perspective of condensed matter physics, polymer physics is originally a branch of statistical physics. Polymer physics and polymer chemistry are also related with the field of polymer science, where this is considered the applicative part of polymers. Polymers are large molecules and thus are very complicated for solving using a deterministic method. Yet, statistical approaches can yield results and are often pertinent, since large polymers (i.e., polymers with a large number of monomers) are describable efficiently in the thermodynamic limit of infinitely many monomers (although the actual size is clearly finite). Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics and dynamics. As a corollary, temperature strongly affects the physical behavior of polymers in solution, causing phase transitions, melts, and so on. The statistical approach for polymer physics is based on an analogy between a polymer and either a Brownian motion, or other type of a random walk, the self-avoiding walk. The simplest possible polymer model is presented by the ideal chain, corresponding to a simple random walk. Experimental approaches for characterizing polymers are also common, using Polymer characterizationmethods, such as size exclusion chromatography, Viscometry, Dynamic light scattering, and Automatic Continuous Online Monitoring of Polymerization Reactions (ACOMP)[5][6] for determining the chemical, physical, and material properties of polymers.
    [Show full text]