Dynamics of a Single Polymer Chain in Gas Phase

Transcription

1 Advanced Studies in Theoretical Physics Vol. 3, 09, no., - 9 HIKARI Ltd, Dynamics of a Single Polymer Chain in Gas Phase Sayed Hasan Department of Physics University of Dhaka, Bangladesh Copyright c 09 Sayed Hasan. This article is distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. Astract As the mean free path is very large in a gas, the collisions the gas particles make with the monomers are allistic. The random force terms in Langevin equation will e correlated due to this allistic nature of collisions. Considering dynamics at theta temperature, this correlation of random forces is otained using Gaussian distriution of a single chain. Susequently this correlation is exploited to find the relaxation time in gas phase. PACS: g Keywords: mean free path, Langevin equation, random force, relaxation time Introduction Modern ionization techniques such as MALDI (Matrix Assisted Laser Desorption/Ionization) [,[,[3 or electrospray ionization [4 have made it possile to have large polymers in gases or in a vacuum. Although much is known aout how polymers ehave in solutions or in melts, there is little information aout polymers in gases. Gas phase studies of polymers are very interesting from a statistical physics point of view, and these studies may also contriute valuale knowledge to the iological and materials sciences. Many fundamental properties of iomolecules can e determined in a gas phase structure. To

2 Sayed Hasan do its task, a iomolecule must have the correct three dimensional conformation. Solvent plays a very important role in determining the conformation and, gas phase studies are helpful in the study of solvent effects. For example, it is possile to oserve hydration effects y adding water molecules one at a time [5. Gas phase experiments are also used to investigate the mechanism of peptide folding [6 and to determine the relative strengths of intramolecular electrostatic interactions and solute-solvent intermolecular interactions. Protein misfolding prolems [7 or comparison to theoretical predictions of DA structure can e studied in the gas phase. Gas phase studies are important for synthetic polymers as well. In most of the gas phase experiments, the drift time of the ionized polymer is measured. This ion moility measurement gives collision cross-sections and information aout conformations are extracted from this. But for large macromolecules their internal dynamics should e considered. Also the dynamics of macromolecules in gases may e useful to oserve sutle features of many reaction mechanisms that are not possile in solutions. For gas phase dynamics, the first thing to consider is the correct equation of motion. Consider the Langevin equation for the Rouse model [8: ζ R n t = k R n n + f n. () This is true in hydrodynamic limit. Although the mean free path is very large in a gas, we can still consider this equation, ut the correlation of random forces would e different [9. Since the mean free path, λ, is large, after hitting a monomer n, a gas particle can strike other monomers without making collisions with other gas particles. So the collisions the gas particles make with the monomers are allistic, as shown in Figure. The allistic nature of collisions implies that random forces on different monomers are correlated, and this correlation depends on the conformations of the polymer. Considering dynamics at θ temperature, we know that the polymer ehaves ideally at this temperature and its conformation is descried y a Gaussian distriution. A particular conformation gives the relative position etween monomers in 3D space. So the impact of allistic collisions would heavily depend on polymer conformation. Correlation of random forces in gas phase dynamics We can think of a gas particle which, after making a collision with some monomer, recoils along some direction, say the X direction. What is the proaility that it will met other monomers efore returning to amient gas?

3 Dynamics of a single polymer chain in gas phase 3 Figure : Gas particles (rown) making allistic collisions with the monomers (lack). Since the mean free path in a gas is large, there is hardly any chance that two gas particles will meet within the pervaded volume of a polymer. Because an ideal chain follows the distriution of a 3D random walk, some properties of this random walker need to e considered to find this proaility. Let P D (, x) e the proaility distriution in one dimension: P D (, x) = π x exp ( x ). x The proaility distriution along the y coordinate, i.e., P D (, y) would e independent of the value previously assigned to x if is large and x is much smaller than the full extension of the chain, i.e., >> and x << [0. Similarly, P D (, z) is independent of x and y when >> and x <<, y <<. So for small extensions of a long chain, P D (, x) depends only on x, P D (, y) only on y, and P D (, z) only on z. We can write the 3D proaility distriution function as the product of three one-dimensional distriutions [0: P 3D (, R)dR x dr y dr z = P D (, R x ) dr x P D (, R y ) dr y P D (, R z ) dr z. If the gas particle is moving along the X direction, we need to consider the random walk of the polymer in the Y Z plane. If we know the proaility that a random walker moving in the Y Z plane will return to the origin, this would give the proaility of meeting other monomers as the gas particle moves along the X direction. From Póly s theorem, we know that a random walk in D is recurrent, and the numer of times that the random walker comes ack to the origin diverges logarithmically. So the proaility to meet other monomers while moving along the X direction approaches unity with increasing monomer numer. Since the distriution of a 3D random walker is homogeneous in space, this would e true for all gas particles regardless of

4 4 Sayed Hasan their recoil direction. The distriution will not e homogeneous for monomers located on the edges, ut that numer is small and we will assume an isotropic distriution for all monomers. Turning ack to the particle that was moving along the X direction, if it just made a collision with monomer n, what is the proaility that it will hit monomer n, escaping all other monomers in etween? This would correspond to the proaility of returning to the origin for the first time in the Y Z plane after taking n n steps. Therefore, we need to know the first-passage time for D random walk. The proaility to return to the origin for the first time in D is [ P nn = 4π ( ) R n n (ln n n ), () where R is the monomer radius and is the ond length and Equation holds when n and n are not very close and we will assume it to e true for n n >. Thus, the random forces on n and n will e correlated with this proaility. We want to solve the Langevin equation for the following normal modes: X p = cos (pπn ) Rn, f p = cos ( pπn) fn, where R n is the position vector of monomer n and f n is the random force on monomer n. The correlation etween random forces of mode p and q is given y (pπn) f p (t) f q (0) = 4 cos fn (t) cos ( qπn ) fn (0). (3) The entire collision process can e divided into two parts. A gas particle may return directly to the surroundings after colliding with a monomer, or it may go from one monomer to other with proaility P nn.ow we will consider the polymer as monomers distriuted within a sphere as in Figure, and then the first process (i.e., when gas particles go ack directly to the environment) can e treated as external interaction for the sphere. Here momentum is transferred to or from the sphere y external forces, and so the correlation of random forces would give the delta term f n f n δ nn. In the second process, the gas particles make successive collisions within the sphere. These involve momentum transfer from monomer to monomer only and no external forces are involved. So we will assume f n = f n to ensure momentum conservation. As f p = cos ( ) pπn fn, there would e a term ( cos( pπn pπn ) cos( )) in the summation for f p. This term puts some restrictions on the correlation of the normal modes, i.e., the normal mode of the friction constant would e different from the normal mode of the friction constant otained from uncorrelated random forces.

5 Dynamics of a single polymer chain in gas phase 5 p 7 p f f 4 f p 9 f 3 p 4 p Figure : Interaction of gas particles with a sphere within which monomers are distriuted. The entire collision process can e divided into two parts. In one part, momentum can e transferred to or removed from the sphere. In another part, there would e momentum transfer from monomer to monomer mediated y gas particles. Here p, p 7, p... are some random amount of momentum. 3 Calculation of relaxation times We can write Equation 3 as f p (t) f q (0) = 4 n>n + 4 [ cos ( qπn dndn cos ( pπn ) (qπn ) cos fn (t) f n (0) }{{ ) } ( ζk B T δ nn δ(t) [ dndn P nn cos ( pπn) (pπn ) cos ) (qπn ) cos f n (t) f n (0). }{{ ) } ( ζk B T δ(t) Consider the summation involving P nn, I(p, q) = [ P nn cos( pπn [ ) cos(pπn ) cos( qπn ) cos(qπn ) n,n = π ( R ) [ sin ( p (n n ) ) sin ( q (n n ) ) [sin ( p (n + n ) ) sin ( q (n + n ) ), n n (ln n n ) where we have used the notation p = pπ and q = qπ. For p = q: I(p) = ( ) R [ sin ( p (n n ) ) sin ( p (n + n ) ). (5) π n n (ln n n ) (4)

6 6 Sayed Hasan To perform the summation we will make a change in coordinates: u = n n, v = n + n. With the new coordinates Equation 5 ecomes [9 I(p) = π ( ) [ R du sin p u u(ln u) u u dvsin p v+ du sin p u u (ln u ) First we will integrate over v for a constant u [9: I(p) = ( ) [ R du sin p u π u(ln u) du sin p u (ln u) + ( ) [ R du sinp u π p u(ln u) du sin4p u. 4p u(ln u) +u We will consider the limiting cases for oth very large and very small p. For large p, the sine function will oscillate very rapidly. Since the last two terms on the right side of Equation 6 are proportional to sin p, they will contriute little. Also, the term in front of the integrals further decreases the contriution of p the last two terms, which can e neglected. For small p, sin p u p u, and the leading contriution from the last two terms on the right side of Equation 6 would e proportional to (p u) 3, so that we can neglect those terms in this limit as well. Therefore, for oth limits, we need to retain only the first two terms in Equation 6. Thus, I(p) π ( R Comining Equations 4 and 7 gives ) [ du sin p u u(ln u) du sin p u (ln u) f p (t) f p (0) = 4 ( ζk B T δ(t) ) dndn cos ( pπn) (pπn ) cos δnn }{{} 4 ( ζk B T δ(t) ) π ( ) [ R u (6). (7) du sin p u u(ln u) du sin p u (ln u) (8) The correlation of random forces would e proportional to the friction constant of mode p, i.e., f p (t) f p (0) = ζ p k B T δ(t). Comining this with Equation 8 gives ζ p = ζ 8ζ π We will write ζ p as ( ) [ R du sin p u u(ln u) du sin p u. (9) (ln u) ζ p = ζ[, (0) dvsin p v..

7 Dynamics of a single polymer chain in gas phase 7 where = 4 π ( ) [ R sin p u u(ln u) sin p u. () (ln u) The relaxation time for mode p is τ p = ζp k p, where k p = 6π k B T p. This is the spring constant for a Gaussian chain. From Equation 0 we will have k p = τ p ζ( ) = ζ k p ( ). Here ζ k p is the Rouse relaxation time: τ pr = ζ k p. Thus τ p = τ pr ( ) )( τ pr ) = + ( τ pr ( ) = + τ pr τ pr ( ) = τ pr + τ pm, () where τ pm = τ pr ( ). Here su-script m is used to indicate that this relaxation arises from multiple collisions of a single gas particle with different monomers. For small p: = 4 π = πp ( ) [ R p ( ) [ R pπ (p u) u(ln u) u (ln u) p (p u) (ln u) pπ u. (3) (ln u) We can make some rough approximations of the term from Equation 3. First consider the sum pπ u. Here ln u saturates quickly and we have an (ln u) integration over u with upper limit. Similarly in the second sum, we have an integration over u with the same upper limit ut there is also a factor in front of the sum. From these we can predict that would e one or two orders of magnitude smaller than [9. This can e considered as a multi relaxation process. We know that the dynamics of polymers in solution also involves different relaxation times, such

8 8 Sayed Hasan as translational and rotational/internal relaxations [, [3. In dilute solutions, dynamic light scattering experiments can resolve translational and internal modes. First the intensity autocorrelation can e measured for different scattering vectors, followed y an inverse Laplace transform to separate these modes. Similarly, dynamic light scattering measurements can e used in gases to determine τ pm. So far we have considered chains at the theta temperature. Aove this temperature, the polymer size would e larger and the polymer would e more porous to the gas particles. Below the theta temperature, the monomer density will increase and the impact of multicollision processes would e larger. 4 Conclusion In summary, we have considered the dynamics of a flexile chain at the theta temperature in the gas phase. For this collision, the random force terms in Langevin equation are correlated, as descried y the statistical property of a Gaussian chain. The relaxation time can e divided into two parts: τ pr and τ pm. Here τ pr is similar to the Rouse relaxation time, ut with the important difference that ζ here is the gas friction constant which is much smaller than the friction constant in solution. The τ pm is comparale to τ pr, although it is larger than τ pr and does not have an exact power law dependence on as is true in the Rouse case. Measurement of the dynamic structure factor can reveal these relaxation times. Acknowledgements. The author wish to thank Prof. Sergei Oukhov who provided valuale advice and comments to the undertaking of the research summarised here. References [ M. Karas, D. Bachmann, U. Bahr and F. Hillenkamp, Matrix-assisted ultraviolet laser desorption of non-volatile compounds, Int. J. Mass Spectrom. Ion Proc., 78 (987), [ K. Tanaka, H. Waki, Y. Ido, S. Akita, Y. Yoshida and T. Yoshida, Protein and Polymer Analyses up to m/z y Laser Ionization Time-of- Flight Mass Spectrometry, Rapid Communications in Mass Spectrometry, (988), [3 F. Hillenkamp, MALDI MS: Practical Guide to Instrumentation, Methods and Applications, Wiley, ew York, 007.

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