CHAPTER 9 AN EXAMPLE OF THE MONTECARLO SIMULATIONS: THE LANGEVIN DYNAMICS Objectves After completng the readng of ths chapter, you wll be able to: Construct a polymer chan. Devse a smple MC smulaton program for a polymer chan. Calculate the end-to-end vector and the radus of gyraton of a polymer chan. Verfy the constancy of the bond segment vector, a key requrementof any polymer chan under equlbrum. Vsualze the moton of the polymer chan through aseres of snapshots. Keywords Polymer chan, Harmonc potental, Langevn dynamcs, End-to-end vector, Radus of gyraton, Bond segment vector. 9.1. Introducton A polymer s a substance comprsng of many repettve structural unts. Examples of polymerc substances are abundant around us. The nonmetallc parts of your computer are made from phenol-formaldehyde polymers, the body of the ball pont pen you use s made from polyethylene, all synthetc clothes are made through polymerzaton processes of sutable raw materals, thedsposable plastc cutleres you use for eatng food are polymerc substances. In bologcal context, essentally all bologcal macromolecules e.g., protens, lpds, polysacchardes are polymerc. Many of the household tems (buckets, mugs, mlk cans, vegetable baskets etc) are also made from polymerc substances. 9.. A smple model for a lnear polymer As polymers comprse of monomers, modelng the polymers may be smplfed by choosng approprate defnton and propertes of the monomers. To llustrate the pont, let us consder a 1
lnear polymer, polyethylene, CH 3 -(CH ) n -CH 3, as an example. Here, we may consder CH 3 and CH as unted atom groups (or beads ) connected through bonds (wth known force constants). The bonds actually behave as sprngs allowng small stretchng and compresson due to vbratonal moton. Thus, smply put, a lnear polymer (lke polyethylene) s modeled as a beadsprng chan. In the bead-sprng chan, the bead partcles nteract through a harmonc potental (known as the Fraenkel potental).for the sake of smplcty, we neglect the bond angle bendng and dhedral torsonal modes of the chan. The Fraenkelpotental has the form U H = ( r ) (9.1) F Fraenkel( r) b0 where,h F s the force constant and b 0 s the equlbrum bead-bead dstance. The Langevn equaton s the most approprate equaton of moton of the beads of the chan. In T the Langevn dynamcs method, moton of each bead of the chan s dctated by a total force F on the bead, whch comprses of force arsng from potental, frctonal force and a random force... T C F R = mr = F + F F F + (9.) where, m s the mass of the bead. The force arsng from the chosen Fraenkel potental are C F ncluded n F. The frctonal force actng on the bead s, F = ξv, where v s the velocty of the bead and ξ s the frcton coeffcent. The frcton coeffcent ξ s related to the fluctuatons of R the random force F through the fluctuaton-dsspaton theorem [See Statstcal Mechancs byd. A. McQuare, Harper and Row, New York; 1976]. F ( t) = 0 (9.3a) F R R R ' ( t) F ( t ) = 6k Tξδ δ ( t t ' ) j B j (9.3b) The temperaturet of the system s mantaned through eq. (9.3), and k B s the Boltzmann constant.in the Monte Carlo smulaton of the Langevn equaton of a polymer chan, we replace the contnuous tme varable wth a small tme step t. If the poston of the th bead at tme step t s denoted by r (t), the smulaton form of the Langevn equaton for the bead can be wrtten as, d T r ( t + t) ) = r ( t) + F ( t) + d ( t) (9.4) k T B where, F T (t) s the total force on the th bead arsng out of the nteracton potental. The random step vector d (t) s characterzed by the two moments, ( t) = 0 d (9.5a)
d ( t) d ( t ' ) = d Iδ δ ( t t ' ) j j (9.5b) where, I s an unt tensor. The random dsplacement d, and tme step t, are related through the dffuson constant D, of the chan. d kbt t = D t = (9.6) ξ It s to be noted that n eq. (9.4) and (9.6), mass of the bead does not appear explctly as t k T does n eq. (9.); but the bead mass s mplct n the frcton coeffcent as, ξ = B = mγ, D where γ s the collson frequency [See Molecular Modellng: Prncples and Applcatons, by A. R. Leach, Second edton, Pearson Educaton Lmted, Essex; 001]. The Langevn dynamcs method s therefore a specal Monte Carlo method, snce the postons of the partcles are mplctly decded by random numbers (see eqs. 9.4-9.6). 9.. Propertes of nterest n smulatng a lnear polymer Any polymer chan s characterzed by several propertes, vz., the end-to-end dstance, ts radus of gyraton and the bond-segment dstance. We defne these quanttes below. 9..1. End-to-end dstance An deal polymer chan s a freely-jonted chan of N monomers. If the postons of the monomers are (r 1, r,, r N ), then the end-to-end vector,rr may be defned as RR = (rr NN rr ) 1 (9.7) The end-to-end dstance, R s the norm of the end-to-end vector, RR. RR = RR. RR RR = RR (9.8a) (9.8b) The two ends of the polymer chan are never concdent, even f the chan forms a col under specal condtons (lke changng a solvent, changng temperature etc). The magntude of the end-to-end dstance, R s always taken to be statstcally averaged n real systems. In the smulatons, R s easly evaluated from the poston coordnates of the end monomers, from whch R can be easly calculated. 9... Radus of gyraton 3
The radus of gyraton, R g of the polymer chan s defned through ts square, RR gg = 1 NN (rr NN =1 rr com ) (9.9) Thus,RR gg = RR gg (9.10) where r are the poston coordnates of the th monomer and r com are the coordnates of the centre-of-mass of the chan. Durng the moton of the chan, one can calculater com and consequently, R g and R g can be easly evaluated. 9..3. Bond segment dstance The bond segment dstance n a polymer chan s a representatve of the so-called Kuhn segment or Rouse segment of the chan. The bond segment dstance s calculated through ts square, bb = 1 NN 1(rr NN 1 =1 +1 rr ) (9.11) So that, bb = bb (9.1) Ths quantty s very mportant and ts magntude must not change n all equlbrum smulatons. Throughout ths chapter, we report the results usng the followng unts: length n terms ofthe bead dameter σand tme as the Monte Carlo steps. 9.3. Smulaton of the lnear polymer For the sake of smplcty, we llustrate the Langevn dynamcs smulaton by consderng a sngle chan of N = 0 beads. Intal confguraton of the chan of N beads has been generated as follows. Takng the frst bead as the seed (placed arbtrarly at the orgn), coordnates (x, y, z ) of the successve beads are generated usng ( x = x 1 + σ snθ cosφ ; y = y 1 + σ snθ snφ ; z = z 1 + σ cosθ ; for =, 3,, N); angles θ and φ are obtaned from, θ = π qq and φ = π * ff respectvely, where qq and ff are two random numbers generated for each bead. Consecutve nter-bead dstances are matched wth σ, before acceptng the coordnates of the new bead to buld the chan. Fgure 9.1 represents the ntal confguraton of the chan. Fgure 9.1: Intal confguraton of a 0-beads chan. 4
Ths chan s acted upon by the Fraenkel potental only [eq. (9.1)] and ts moton s followed for 10 5 MC steps. Durng the smulaton, the square of end-to-end vector R and the square of the radus of gyraton R g are calculated along wth thesquare of the average bond-segment dstance<b >. 9.4. Results 9.4.1. Square of the end-to-end vector R, square of the radus of gyraton R g and square of the average bond-segment dstance <b >. Fgures 9., 9.3 and 9.4 present the varatons of thesquare of the end-to-end vector R, square of the radus of gyraton R g and square of the average bond-segment dstance <b >, respectvely, wth tme steps of the MC smulaton. 3.3 0 beads polymer chan 3. R /σ 3.1 3.0.9 10 0 10 1 10 10 3 10 4 10 5 t (MC steps) Fgure 9.: Varaton of thesquare of the end-to-end vector,r of the 0 beads polymer chan. 5
0.36 0.35 0 beads polymer chan 0.34 R g /σ 0.33 0.3 0.31 0.30 10 0 10 1 10 10 3 10 4 10 5 t (MC steps) Fgure 9.3: Varaton of thesquare of the radus of gyraton,r g of the 0 beads polymer chan. 1.00 0 beads polymer chan 1.001 <b >/σ 1.000 0.999 0.998 10 0 10 1 10 10 3 10 4 10 5 t (MC steps) Fgure 9.4: Varaton of thesquare of the average bond segment dstance,<b > of the 0 beads polymer chan. Note the constancy of <b >. As ther andr g are equlbrum propertes of the polymer chan, these are expected to change very lttle wth tme. The slght varatons seen n Fgs 9. and 9.3, after ca. 10 4 MC steps are ndcatons of the chan to start colng. In the case of the square of the average bond-segment dstance <b >, the magntude should not change at all and ths s what s revealed n Fg. 9.4. 6
9.4.. Snapshots of the system at dfferent tmes Snapshots of the polymer chan durng the 10 5 MC steps smulaton have been shown n Fgure 9.5. Confguraton at step no 100 Confguraton at step no 1000 Confguraton at step no 0000 Confguraton at step no 5000 Fgure 9.5: Snapshots of the polymer chan durng the 10 5 MC steps usng the Fraenkel potental, gnorng the bond angle bendng and dhedral torsonal modes of moton. Note that after ca. 10 4 MC steps, the chan starts showng a tendency to form cols. Questons 1. What changes would you expect n the chan f the angle bendng potental and a torsonal potental were consdered along wth the Fraenkel potental? 7
. The temperature enters nto the smulaton through eq. (9.3). In case the temperature needs to be changed, whch parameters get affected? 3. What s the orgn of the constancy of <b >? 4. In a system wth 100 chans of polyethylene (each chan s of 0 beads), do you expect varatons n R,R g and<b >? Justfy wth reasons. 8