Polymer Journal, Vol. 39, No. 4, pp. 359–368 (2007) #2007 The Society of Polymer Science, Japan Normal Modes and Their Dispersion in Form of Nylon-6 ( NY6) Sandeep Kumar SHUKLA, Naresh KUMAR, Abhishek Kumar MISHRA, y Poonam TANDON , and Vishwambhar Dayal GUPTA Physics Department, Lucknow University, Lucknow 226 007, India (Received August 28, 2006; Accepted January 12, 2007; Published March 5, 2007) ABSTRACT: Nylon-6 [-(CH2)5–NHCO-]n ( NY6) is a synthetic polymer widely used in industrial applications. It has a planar zigzag conformation. A comprehensive study of the normal modes and their dispersion in NY6 using Urey Bradley force field is being reported. Crossing between the various pairs of modes of dispersion curves have been explained as arising due to internal symmetry in the energy momentum space. The heat capacity derived from the dis- persion curves via the density-of-states, is in good agreement with the experimental measurements obtained from the Athas data bank. [doi:10.1295/polymj.PJ2006103] KEY WORDS Nylon-6 / -Form / Phonon Dispersion / Heat Capacity / Density-of-States / The word Nylon has been accepted as generic term the microscopic behaviour of the long chain molecule for synthetic polyamides, Nylon-6 [-(CH2)5–NHCO-]n with the macroscopic properties such as entropy, en- is one of the most important members of this family. It thalpy, specific heat etc. is highly useful for industrial applications because of We present here a complete normal mode analysis its high tensile and impact strength, firm stability at of NY6 with phonon dispersion in the first Brillouin high temperature, good abrasion resistance and self- Zone using Urey Bradley force field (UBFF).3,4 This lubricating properties. It also retains both tough and polymer has been subjected to several spectroscopic flexible at low temperatures. Commercially it can be studies (IR, Raman and INS)5–12 by several workers. processed by conventional processing methods such Tadokoro et al.11 have reported normal mode analysis as injection molding, extrusion blow molding, and of NY6 considering methylene group as a point mass special grade of Nylon-6 are available for rotational thereby neglecting the interaction between carbon and molding and thermoforming. hydrogen atoms where as Jakes and Krimm5 have Nylon-6 crystallizes in two different forms namely used simple valence force field in their normal mode and forms.1,2 Thermodynamically stable form calculations. Our calculations are based on UBFF, takes on planar zigzag conformation and the molecu- which in addition to valence force field accounts for lar chains are arranged in sheets by means of hydro- the non-bonded interactions in the gem and cis config- gen bonding between antiparallel chains, while the uration and the tension terms. In this force field the form takes a helical symmetry with the secondary potential energy expression does not have quadratic amide group at 66 with respect to the plane of CH2 cross terms. The force constants are supplemented zigzag and sheets of parallel chains are joined by hy- by the repulsive forces between non-bonded atoms, drogen bonds. In both the forms the neighbouring which simulate the van der Waals force13 between chains are arranged in sheets of parallel but oppo- them. It gives a better description of intra and inter sitely directed molecules. Lattice parameters along unit interactions, and arbitrariness in choosing the the polymer axis are slightly shorter (b ¼ 16:88 A˚ )2 force constants is reduced, thereby enabling us to ar- in the form as comparison to that of the form rive at a better unique force field. (b ¼ 17:24 A˚ ).1 Vibrational spectroscopy plays an important role in THEORY the elucidation of polymeric structure. Normal mode analysis besides identification of various modes pro- Calculation of Normal Mode Frequencies vides an insight into Infrared absorption (IR), Raman Normal mode calculation for a polymeric chain spectra and Inelastic Neutron Scattering (INS). An was carried out using Wilson’s GF matrix method14 overall understanding of vibrational dynamics in a as modified by Higgs15 for an infinite polymeric chain. polymer involves calculation of the dispersion curves. The vibrational secular equation to be solved is These curves provide knowledge of degree of uninter- jGðÞFðÞÀðÞIj¼00 ð1Þ rupted sequence lengths in an ordered conformation. The dispersion curves also facilitate correlation of where is the phase difference between the modes of yTo whom correspondence should be addressed (E-mail: poonam [email protected]). 359 S. K. SHUKLA et al. adjacent chemical units, GðÞ is the inverse kinetic Table I. Internal coordinates and force energy matrix and FðÞ is the force field matrix for a constants for NY6 (mydn/A˚ ) À1 certain phase value. The wavenumber viðÞ in cm Internal Force Internal Force 2 2 2 are related to eigen values by iðÞ¼4 c ½viðÞ . Coordinates Constants Coordinates Constants A plot of viðÞ versus gives the dispersion curve (C=O) 7.85 ’ (C4-C5-C6) 0.470 (.600) for the ith mode. The use of the type of force field (C1-C2) 2.85 ’ (H-C5-C6) 0.470 (.220) is generally a matter of one’s chemical experience (C2-H) 4.18 ’ (C5-C6-N) 0.150 (.600) 16 and intuition. In the present work, we have used (C2-C3) 3.24 ’ (C5-C6-H) 0.440 (.220) Urey-Bradley force field as it is more comprehensive (C-H) 4.18 ’ (H-C6-H) 0.362 (.360) then valence force field. The Urey-Bradley takes into (C-C) 2.98 ’ (H-C-N) 0.245 (.780) account both bonded and non-bonded interactions as (C5-C6) 3.28 ’ (C6-N-H) 0.292 (.520) (C -H) 3.91 ’ (C-N-C) 0.440 (.540) well as internal tensions. Potential energy for this 6 (C -N) 2.35 ’ (H-N-C ) 0.453 (.520) force field can be written as 6 1 X (N-H) 5.38 ’ (N-C=O) 0.890 (.900) 0 ðmÞ ðmÞ ðmÞ 2 (N-C1) 5.75 ’ (N-C1-C2) 0.400 (.600) V ¼ Kj;krj;k ðÁrj;k ÞþKj;kðÁrj;k Þ =2 ’ (O=C -C ) 0.890 (.900) ! (N-H) 0.165 m;j;Xk 1 2 ’ (C1-C2-H) 0.406 (.215) ! (C=O) 0.519 þ H0 rðmÞrðmÞðÁ ðmÞ Þ i;j;k i;j j;k i;j;k ’ (H-C2-H) 0.389 (.340) (C1-C2) 0.010 m;i;j;k ’ (H-C2-C3) 0.410 (.215) (C2-C3) 0.011 ðmÞ ðmÞ 2 þ H r ðÁ Þ =2 ’ (C1-C2-C3) 0.750 (.500) (C3-C4) 0.009 Xi;j;k j;k i;j;k 0 ðmÞ ðmÞ ðmÞ 2 ’ (C2-C3-H) 0.440 (.230) (C4-C5) 0.019 þ F q ðÁq ÞþFi;kðÁq Þ =2 i;k i;k i;k i;k ’ (H-C-C) 0.458 (.230) (C5-C6) 0.032 m;i;j;k X X ’ (C2-C3-C4) 0.480 (.500) (C6-N) 0.011 2 ! 2 þ Kj ðÁjÞ þ Kj ðÁ!jÞ ð2Þ ’ (H-C-H) 0.392 (.340) (N-C1) 0.030 j j ’ (C3-C4-C5) 0.470 (.600) where the symbols have their usual meaning. The Note: 1. , ’, ! and denote stretch, angle bend, wag and torsion primed quantities are introduced as internal tensions. respectively. 2. Non-bonded force constants are given in parentheses. Non-bonded interactions involve attraction and repul- sion of atoms due to the overlap of their electron shells. These effects are usually expressed by the 6- the corrections to F0 so that the errors Á are mini- exp or 6-12 type potentials. The tension terms are as- mized. We used the theory of least squares and calcu- sumed to be all zero. late Recently, spectroscopically effective molecular me- J0PÁ ¼ðJ0PJÞÁF ð5Þ chanics models have been used for inter and intra mo- lecular interactions consisting of charges, atomic di- where P is the weighting matrix and J0 is the transpo- poles and van der Waals (non bonded) interactions.13 sition of J. The solution of this equation is obtained by The force constants, including those for the interac- inverting J0PJ to give tion of first and third non-bonded atoms, which give ÁF ¼ðJ0PJÞÀ1J0PÁ ð6Þ the ‘‘best fit’’, are given in the Table I and have been obtained by least squares fitting. In order to obtain the If the number of frequencies is greater than the ‘‘best fit’’ with the observed wave numbers the follow- number of F matrix elements, the matrix J0PJ should ing procedure is adopted. be non-singular and be obtain the corrections ÁF, Force constants were initially transferred from the which will minimize the sum of the weighted squares molecules (PCL17 and poly (L-valine)18 in the pres- of the residuals. This minimum sum provides the ent case) having similar groups placed in the similar ‘‘best fit’’. If the corrections ÁF are fairly large, the environment. Thus starting with the approximate F linear relation between force constant and frequency matrix Fo and observed frequencies obs (related term in the matrix eq 3 breaks down. In such a situa- through a constant), one can solve the secular matrix tion, further refinement using higher order terms in equation: the Taylor’s series expansion of Ái is needed. King et al.19 developed this procedure. GF0L0 ¼ L00 ð3Þ Let Ái ¼ iobs À i0 in the above equation. It can be Calculation of Specific Heat shown that in the direct order of approximation Dispersion curves can be used to calculate the spe- cific heat of a polymeric system. For a one-dimension- Á ¼ JÁF ð4Þ al system the density of state function or the frequency where J is computed from L0.