A computer simulation model of electrodeposition of polymer chains on an impenetrable wall is used to evaluate the power-law scaling exponents (νx(y)) for the longitudinal and transverse spread, Rgx(y)∼Lcνx(y); we find that the exponents  νx(y) depend on the field strength, i.e., they are nonuniversal. A conformational crossover is observed for the transverse spread from the bulk with νy&sime;1/3-2/3 to the wall with νy&sime;2/3-1. A similar crossover also occurs for the longitudinal component of Rg but with an opposite trend, i.e., magnitude of νx is larger in bulk than at the wall

Foo, Grace M.

Pandey, Ras B.

English

Aquila Digital Community

The University of Southern MississippiThe Aquila Digital CommunityFaculty Publications10-13-1997Nonuniversal Scaling and ConformationalCrossover of Polymer Chains in an ElectrophoreticDepositionGrace M. FooNational University of SingaporeRas B. PandeyUniversity of Southern Mississippi, ras.pandey@usm.eduFollow this and additional works at: http://aquila.usm.edu/fac_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Faculty Publications byan authorized administrator of The Aquila Digital Community. For more information, please contact Joshua.Cromwell@usm.edu.Recommended CitationFoo, G. M., Pandey, R. B. (1997). Nonuniversal Scaling and Conformational Crossover of Polymer Chains in an ElectrophoreticDeposition. Physical Review Letters, 79(15), 2903-2906.Available at: http://aquila.usm.edu/fac_pubs/5434VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997Nonuniversal Scaling and Conformational Crossover of Polymer Chainsin an Electrophoretic DepositionGrace M. Foo1 and R. B. Pandey21Supercomputing and Visualization Unit, National University of Singapore, Singapore 1192602Department of Physics and Astronomy, Program in Scientific Computing, University of Southern Mississippi,Hattiesburg, Mississippi 39406-5046(Received 26 June 1997)A computer simulation model of electrodeposition of polymer chains on an impenetrable wallis used to evaluate the power-law scaling exponents (nxs yd) for the longitudinal and transversespread, Rgxs yd , Lnxs ydc ; we find that the exponents nxs yd depend on the field strength, i.e., they arenonuniversal. A conformational crossover is observed for the transverse spread from the bulk withny . 1y3 2y3 to the wall with ny . 2y3 1. A similar crossover also occurs for the longitudinalcomponent of Rg but with an opposite trend, i.e., magnitude of nx is larger in bulk than at the wall.[S0031-9007(97)04202-6]PACS numbers: 61.41.+e, 81.15.PqDynamics of interfacial growth and scaling describedby the Kardar-Parisi-Zhang (KPZ) model [1] and its con-nection to scaling in directed polymers [2] has opened upenormous possibilities for addressing interesting questionsin a variety of systems for about a decade [3–5]. Kardarand Zhang [2] made a fundamental contribution to the scal-ing of transverse fluctuations with the length of directedpolymers subject to external impurities, i.e., polyelec-trolytes in a gel matrix. They argue that a configura-tion of directed polymer can be regarded as the world ofline. Therefore, a path integral representation of the over-all Boltzmann weight WsRy , Nd can be used to describe thetransverse fluctuation Rgy ( y component of the radius ofgyration) with the length N of the x-directed polymer, i.e.,W sRy, Nd ­Z sRy ,Nds0,0dDR0ysN0d exp(2Z N0dN 0fsgy2d sdR0yydNd2 1 msR0y , N0dg), (1)where g is a bare line tension, and msRy , Nd is an impu-rity term with fluctuation kmsx, Ndmsx0, N 0dl ­ s2dsN 2N 0ddd21sRy 2 R0yd in d-dimensional space. In two di-mensions, with the impurities, the scaling exponent n forthe transverse fluctuation with the length Ry , Nny , ny ­2y3 exactly. Mapping Eq. (1) to the randomly stirredBurgers equation and using dynamical renormalizationgroup method, and by numerical simulations in d $ 2,Kardar and Zhang predicted [2,6,7] that the scaling ex-ponent ny is superuniversal with ny ­ 2y3 in d ­ 2, 3,and 4.We consider a computer simulation model for electro-deposition of polymer chains [8] to study the universalityof scaling exponent of the chains [2]. Further we addressthe following more general question: How does the con-formation of chain change from bulk to the surface [9–11] as a function of electric field (B) and temperature (T )?We find that the scaling exponent of the transverse (lon-gitudinal) spread of the polymer chains at the wall (bulk)is nonuniversal as it depends on the field strength (B) andtemperature and that its magnitude for the chains at the walldiffers substantially from that in the bulk. Studying theconformation of chains in such a model could be useful inunderstanding the stability and interfacial sliding of poly-mers [12–16] and friction in various applications includinglubrications, spray painting, and coating under pressure.Electrophoretic flow of polymers through a gel matrix,i.e., a gel electrophoresis [17–24] is another examplewhere the knowledge of chains conformations as they de-posit on a wall (pore boundary of gel) could be useful inunderstanding and assessing the errors in such applicationsas DNA fingerprinting.We consider a discrete lattice of size Lx 3 Ly with largeaspect ratios (LxyLy); Lx ­ 1000, Ly ­ 200 is used formost of our data but different lattice sizes are used to checkfor the severe finite size effects. Polymer chains [each oflength Lc with sLc 1 1d nodes connected by a constantbond of unit length] are released at about a constantrate from one end (source end x ­ 1) of the sample andare deposited at impenetrable wall at the opposite end(x ­ Lx) via “slithering snake” (reptation dynamics). Inaddition to excluded volume, a nearest neighbor polymer-polymer repulsive, and polymer-wall attractive (adsorbing)interactions are considered [8]. The external field B iscoupled with the movements of chain nodes via interactionenergy 2BDX, where DX (­ 0 or 61) is the displacementof a node along the x direction. Metropolis algorithm isused to reptate the chains. An attempt to move chainsLx 3 Ly times is defined as unit Monte Carlo step (MCS).Profiles of polymer density and radius of gyrations arestudied in detail. The simulation is carried out for asufficiently long time to obtain stable profiles at the wall0031-9007y97y79(15)y2903(4)$10.00 © 1997 The American Physical Society 2903VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997FIG. 1. Snapshot of chains of length Lc ­ 120 at T ­ 1 andB ­ 2.0.and around in the bulk. We use a sufficiently large numberof independent runs to estimate the average quantitiesreliably.Figure 1 shows a typical snapshot of the instantaneouschain configurations. We see that a sufficiently largeconcentration of polymer chains has already evolved at thewall extending into a considerable fraction of neighboringlayers in the bulk. A typical profile of monomer densityand radius of gyration as a function of x is presented inFig. 2. The density profile suggests that it has reached thestable value at the wall and that it decreases as we go intothe bulk (800 # x , Lx); the decay of density away fromthe wall is primarily due to steric screening and mainly dueto entropic contribution of free energy. Various interestingFIG. 2. Profiles along x direction of (a) monomer density and(b) radius of gyration for chains of length Lc ­ 40 at T ­ 1.Lattice size 1000 3 200 with 20 independent samples is used.effects such as the onset of oscillations near the wall andscaling of polymer density with the chain length and effectsof field has been described in detail elsewhere [8]. Theprofile of the radius of gyration exhibits a contrast invariation of its x and y components near the wall: whileRgx decays with x from bulk to wall, Rgy increases, andthe rate of variation is enhanced on increasing the field.In order to evaluate the scaling exponents nx and nyfor the longitudinal (Rgx , Lnxc ) and transverse (Rgy ,Lnyc ) components of the radius of gyration (Rg , Lnc ), weFIG. 3. A typical (a) Rgx , (b) Rgy , and (c) Rg versus Lc plotfor chains at T ­ 5, with the same statistics as in Fig. 2.2904VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997FIG. 4. Exponent n versus B for chains at T ­ 10.examine their variation with the chain length on a log-log scale for chain at the wall and in the bulk. A typicalvariation is shown in Fig. 3 for various values of fieldstrength. The linear fit seems consistent with the power-law scaling at least with a leading exponent. The lon-gitudinal component in the bulk shows that the exponentnx depends on the field strength; a variation of the expo-nent with field is presented in Fig. 4. It is rather difficultto provide a closed form empirical expression based onour data, nevertheless, a nonmonotonic dependence couldbe speculated from a close inspection. For example, atthe temperature T ­ 10.0, nx increases (nx ­ 0.7 1.0)on increasing the field in low field regime (B ­ 1.0 5.0)before it saturates to its maximum value (nx ­ 1.0) athigh field strength (B $ 5.0); at B # 1.0, nx shows an in-creasing trend with decreasing field leading to a minimumat B . 1.0. The longitudinal component (Rgx) at the wallalso seems to suggest a power-law scaling but with muchlower exponent values (nx . 0.3 0.4 at B ­ 0.5 5.0,T ­ 10.0); the magnitude of the exponent is less sensi-tive to field strength.The transverse spread of the chain, described by Rgy ,also exhibits a power-law scaling both in the bulk as wellas at the wall [see Fig. 3(b)]. The nature of its variationTABLE I. Exponent n in Rg , Lnc . x and y components of Rg are described by exponentsnx and ny , respectively. The error bar Dn in evaluating the exponents varies. Dn ,0.07 0.10 for the chains in the bulk. There are considerably larger error bars (as largeas ,0.15) for the chains at the wall due to poor statistics; many data points do not fit thepower-law dependence indicated by 2. The value of exponents slightly larger than 1 aretruncated to 1 and exponents with relatively larger error bar are identified by p.Temperature Bias Bulk WallT B n nx ny n nx ny100.0 5.0 0.73 0.68 0.77 0.87 0.38 1.00*100.0 10.0 0.69 0.72 0.65 0.70 0.36 0.8210.0 0.3 0.77 0.75 0.77* 0.88 0.37 1.010.0 0.5 0.74 0.77 0.70 0.75 0.32 0.8910.0 1.0 0.68 0.69 0.67 0.70 0.39 0.8110.0 2.0 0.67 0.76 0.54 0.64 0.41 0.7410.0 5.0 0.84 1.00 0.33 0.65 0.50 0.7510.0 10.0 1.00* 1.00* 0.26 0.76 0.43* 1.00*5.0 0.3 0.74 0.74 0.74 0.78 0.39* 0.91*5.0 0.5 0.67 0.70 0.63 0.67 0.43 0.775.0 1.0 0.68 0.79 0.51 0.65 0.46 0.735.0 2.0 0.80 0.98 0.36 0.66 0.47 0.765.0 5.0 1.00 1.00 0.28 0.78 0.40 1.005.0 10.0 1.00 1.00 0.36 0.98* 0.41 —1.0 0.1 0.73 0.76 0.68 0.91* 0.42* 1.00*1.0 0.3 0.82 0.97 0.43 0.90 0.43 1.001.0 0.5 0.94 1.00 0.40 0.91 0.42* 1.001.0 1.0 1.00 1.00 0.39 1.12 0.33* —1.0 2.0 1.00 1.00 0.37 0.96 0.36* —0.5 0.1 0.74 0.84 0.54 1.00* 0.28* —0.5 0.3 0.97 1.00 0.41 1.00* 0.23 —0.5 0.5 1.00 1.00 0.40 — — —0.5 1.0 1.00 1.00 0.33 — 0.11* —0.5 2.0 1.00 1.00 0.46 1.00* 0.15* —0.1 0.1 1.00 1.00 0.46 — — —0.1 0.3 1.00 1.00 0.37 — 0.05* —0.1 0.5 1.00 1.00 0.45 1.00* 0.16 —0.1 1.0 1.00 1.00 0.41 1.00* 0.22 —2905VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997at the wall is different from that in the bulk but with areverse trend in contrast to its longitudinal component.For example, at the temperature T ­ 10.0, the magnitudeof the exponent at the wall ny varies substantially withthe field (ny . 0.7 1.0); it shows a nonmonotonic depen-dence with a minimum, unlike corresponding longitudinalvalue (nx) at the wall but somewhat similar to nx in thebulk [see Fig. 4]. An analogous trend is also seen for thetransverse value of the exponent ny in the bulk. A de-tailed estimate of these exponents at various values of Bin a range of temperature T ­ 0.1 100.0 is presented inTable I. We see that these exponents also depend on thetemperature. For example, at B ­ 1.0, nx in bulk (ny atthe wall) increases from about 0.7 (0.8) on reducing thetemperature from T ­ 100.0 to 1.0; i.e., the field becomesmore effective at low temperatures. The magnitude of theexponent n for the radius of gyration (Rg) has the sameorder of value both at the wall and in the bulk (2y3 at highT to 1 at low T ). The role of the x and y componentsin the bulk are reversed from that at the adsorbing wallwhich is consistent with previous simulations in differentsystems [15,25]. Thus, we find that for our model (1) thepower-law scaling exponent for both the longitudinal andthe transverse spread of the radius of gyration is nonuni-versal as it depends on the field strength and temperature.(2) The crossover in scaling of the longitudinal componentof the radius of gyration from bulk to wall is opposite tothat for the transverse spread. The details including theeffects of the wall interaction on profile and scaling willbe published elsewhere.This work is supported by a NSF-EPSCoR grant.G.M. F. acknowledges support of computer time at theCADCAM and Computer Centre of the National Univer-sity of Singapore.[1] M. Kardar, G. Parisi, and Y.C. Zhang, Phys. Rev. Lett.56, 889 (1986).[2] M. Kardar and Y.C. Zhang, Phys. Rev. Lett. 58, 2087(1987).[3] Dynamics of Fractal Surfaces, edited by F. Family andT. Vicsek (World Scientific, Singapore, 1991).[4] A. L. Barabasi and H. E. Stanley, Fractal Concepts inSurface Growth (Cambridge University Press, Cambridge,1995).[5] R. P. Wool, Polymer Surfaces: Structure and Strength(Hanser Press, New York, 1995).[6] D. A. Huse, C. L. Henley, and D. S. Fisher, Phys. Rev.Lett. 55, 2924 (1985).[7] D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. A16, 732 (1977).[8] G.M. Foo and R. B. Pandey, J. Chem. Phys. (to bepublished).[9] G. J. Fleer et al., Polymers at Interfaces (Chapman andHall, London, 1993).[10] E. Eisenriegler, Polymers near Surfaces (World Scientific,Singapore, 1993).[11] S. A. Safran, Statistical Thermodynamics of Surfaces,Interfaces, and Membranes (Addison-Wesley, Reading,MA, 1994).[12] G. S. Grest, Phys. Rev. Lett. 76, 4979 (1996).[13] U. D’Ortona, J. D. Coninck, J. Koplik, and J. R. Banavar,Phys. Rev. Lett. 74, 928 (1995).[14] R. Zajac and A. Chakrabarti, J. Chem. Phys. 104, 2418(1996).[15] P. Y. Lai, J. Chem. Phys. 103, 5742 (1995).[16] S. K. Kumar, M. Vacatello, and D.Y. Yoon, J. Chem.Phys. 89, 5206 (1988).[17] C. Wu et al., Electrophoresis 17, 1103 (1996).[18] D. A. Hoagland, D. L. Smisek, and D.Y. Chen, Elec-trophoresis 17, 1151 (1996).[19] G.W. Slater and J. Noolandi, Biopolymers 28, 1781(1989).[20] J.M. Deutsch, Science 240, 922 (1988).[21] E. O. Shaffer and M.O. de la Cruz, Macromolecules 22,1351 (1989).[22] L. S. Lerman and H. L. Frisch, Biopolymers 21, 995(1982).[23] O. J. Lumpkin, P. Djardin, and B. H. Zimm, Biopolymers24, 1573 (1985).[24] J. Noolandi et al., Phys. Rev. Lett. 58, 2428 (1987).[25] J. Baschnagel and K. Binder, Macromolecules 28, 6808(1995).2906

Nonuniversal Scaling and Conformational Crossover of Polymer Chains in an Electrophoretic Deposition

https://aquila.usm.edu/cgi/viewcontent.cgi?article=6601&amp;context=fac_pubs

Nonuniversal Scaling and Conformational Crossover of Polymer Chains in an Electrophoretic Deposition

Abstract

Similar works

Full text

Available Versions

Aquila Digital Community