Relaxation time of non-linear polymers in dilute solution via computer simulation

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1 Journal of Non-Crystalline Solids 352 (2006) Relaxation time of non-linear polymers in dilute solution via computer simulation J.G. Hernández Cifre *, R. Pamies, M.C. López Martínez, J. García de la Torre Departamento de Química Física, Facultad de Química, Universidad de Murcia, Murcia, Spain Available online 28 August 2006 Abstract The longest relaxation time of polymer chains with varying topology (linear, regular star and ring) and flexibility (hard and soft connectors) is computed numerically by using two approaches: decay of certain property values after cessation of a perturbation (electric field or flow field) and fit of the time correlation function of some characteristic chain vector under equilibrium. The former approach gives rise to the relaxation time obtained in linear viscoelasticity whereas the latter gives rise to that obtained through dielectric measurements. In order to generate the model chain trajectory we use the Brownian dynamics simulation technique including hydrodynamic interaction and excluded volume. In this work we evaluate and present briefly the influence that the topology and the spring type used to connect the beads of the model have on the values of the longest relaxation time. Ó 2006 Elsevier B.V. All rights reserved. PACS: 83..Mj; Rs; St Keyword: Rheology 1. Introduction In the Rouse Zimm description of the dynamics of polymer chains, the polymer is represented by a beadand-spring chain and the dynamics is essentially determined by the normal modes of the chain [1,2]. For a chain of N beads there are N modes, and a relaxation time, s i,is associated to each of them. The Rouse theory provides analytical solutions to compute relaxation times for the most simplest cases [3], i.e. chain models that do not include effects as fluctuating hydrodynamic interaction (HI), or excluded volume (EV). Thus, for a model with N beads that are arbitrarily connected (with an arbitrary topology), the connectivity is represented by the N N Rouse matrix, A, given by * Corresponding author. Tel.: ; fax: address: jghc@um.es (J.G. Hernández Cifre). 8 >< m i if i ¼ j A ij ¼ 1 if beads i and j are connected >: 0 otherwise where m i is the number of beads to which bead i is attached. The Rouse matrix can be diagonalized to obtain its eigenvalues, k i, and eigenvectors, v i. The relaxation times (when HI is neglected in the Rouse hydrodynamic treatment) are given by s i ¼ f ; ð1þ 2Hk i where f is the friction coefficient of the beads and H is the Hookean constant of the Gaussian springs, given by H =3k B T/b 2 where b 2 is the mean squared spring length. There is a trivial, zero eigenvalue. The so-called first Rouse mode corresponds to the smallest non-zero eigenvalue, k 1. Associate to this mode is the longest relaxation time, s 1. For example, as it is known, when a flexible polymer molecule undergoes elongational flow its property values experience a sudden increase when the elongational rate exceeds /$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:.16/j.jnoncrysol

2 5082 J.G. Hernández Cifre et al. / Journal of Non-Crystalline Solids 352 (2006) certain threshold value, phenomenon known as coil-stretch transition, and the value of that critical elongational rate is ruled by the longest relaxation time of the chain. The eigenvalues and eigenvectors of A for a general structure can be found by direct numerical diagonalization. However, normal modes and relaxation times are greatly influenced by hydrodynamic interaction, which was ignored in the Rouse theory, and included in an approximate (preaveraged) manner in the Zimm theory. By using the Brownian Dynamics technique we can simulate the dynamics of polymers modeled as bead-and-spring chains including HI and EV and from the set of conformations generated we can evaluate the relaxation times. We focus this work to the study of three simple topologies: linear, ring and regular star. Although in the Rouse Zimm treatment there exists a universal series of relaxation times, they appear in two different ways depending on the property that is being observed. Thus, in the treatment of linear viscoelastic properties, the relaxation times are those that we denote as s i (unprimed). The same times take place in the decay of electric birefringence, related to the second order Legendre polynomial of the orientation angle of the bond vectors. However, in other properties the relaxation times differ by just a factor of two [3]. This is the case for dielectric relaxation, and for the decay of the correlation functions of the Rouse modes and dynamic light scattering. These relaxation times will be denoted as s 0 i (primed). The relationship between both relaxation times is simply, at least for flexible polymer chains that follow a Rouse Zimm dynamics and present a Gaussian statistic at equilibrium, s 0 i ¼ 2s i. Two methods to obtain the longest relaxation time are then evaluated. The first one is based on the relaxation decay of some polymer property after cessation of a perturbation. The second one is based on the correlation of the first Rouse mode from a set of equilibrium conformations. The influence of the topology, HI and EV is shown. 2. Theory and methodology The polymer molecule is modeled as a bead-and-spring chain [4] with N beads connected by N 1 springs (in case of linear and star chains) or N springs in case of cyclic topology. Regular stars are then formed by a central bead and F arms, the so-called functionality, of equal length N arm =(N 1)/F. In order to connect neighboring beads, we employed usually FENE (finite extensible non-linear elongational) springs which follow the force law [4] F ðsþ H ¼ Q; ð2þ 2 1 ðq=q max Þ where Q is the spring vector, Q max the maximum spring length and H =3k B T/b 2 the spring constant. Besides, we also used Hookean springs, F ðsþ ¼ HQ 2 ; ð3þ and hard Fraenkel springs [4], F ðsþ ¼ HðQ Q 0 Þ 2 ; ð4þ where Q 0 is the equilibrium connector length and HQ 2 0 is large compared to K B T, say H ¼ 0k B T =Q 2 0. Intramolecular, excluded volume interactions to mimic solvent quality are simulated by introducing interaction forces between non-neighboring beads. An adequate choice to represent these non-bonded interactions is the Lennard- Jones (LJ) potential [5]. The dynamics of the polymer chain is monitored from trajectories of individual molecules obtained by BD simulation. We employ a predictor corrector version of the Ermak and McCammon algorithm proposed by Iniesta and García de la Torre [6]. Simulations with and without HI were performed. Fluctuating hydrodynamic interaction between beads is taken into account using the Rotne Prague Yamakawa tensor [3,7]. For the bead friction we use a Stokes coefficient f =6pg s r, where g s is the solvent viscosity and r = 0.257b, which correspond to a dimensionless HI parameter h * = Along the work, quantities are given in the following unit system: b for length, k B T for energy and fb 2 /k B T for time. For simplicity, we will not use any special notation to indicate dimensionless quantity. In the unit system employed, the Hookean spring constant becomes H = 3. The simulation time step was Dt = 4, as required when the Lennard-Jones potential is present in the computation. In this work, two approaches are employed in order to compute the longest relaxation time of the molecule: decay of a property value after cessation of a perturbation (electric field or flow field) and fit of the time correlation function of some characteristic chain vector under equilibrium Relaxation time from decay after cessation of perturbation A method to obtain the longest relaxation time of the polymer chain consists of perturbing a sample of non-interacting polymer molecules from equilibrium with some external agent, usually a flow field or an electric field (if the molecule is charged) and, once the chains reach the corresponding steady sate, switching the perturbation off. Then conformational relaxation of the polymer chains start to occur up to reach equilibrium again. A multiexponential fit of the time decay of some sample average property, y(t), is performed and the spectrum of relaxation times is obtained from the exponents of the multiexponential fitting equation [8]: yðtþ ¼a 1 e t=s 1 þ a2 e t=s 2 þ, where s1 is the longest relaxation time. Three types of perturbations were checked in this work: a shear flow with velocity field given by v x ¼ _cy (_c the shear rate) and v y = v z = 0, an elongational flow with velocity field given by v x ¼ _ex (_e the elongational rate), v y ¼ 1 _ey 2 and v z ¼ 1 _ez, and finally an electric field. In the latter case 2 we consider a polar chain with dipole moments along the backbone bonds in the polymer or the springs in the model

3 <s 2 > J.G. Hernández Cifre et al. / Journal of Non-Crystalline Solids 352 (2006) <s 2 > I OFF II ON III OFF time Slope ~ -1/τ time Fig. 1. (a) Time evolution of the mean squared radius of gyration for a FENE (Q max = ) chain of N = 13 with HI and no-ev subjected to a shear flow in an off on off experiment. (b) Semilog plot of the decay region of (a). C ee = <r(t 0 )r(t 0 +t)>t (type A in the notation of Stockmayer). Upon application of an electric field, E, only the end beads experience net forces F = qe and F, where q is the charge displacement of the dipoles. Applied perturbation must be intense enough in order to displace significantly molecules from equilibrium but without producing high stretching of the chains and enter in the non-linear viscoelastic regime. In case of using Hookean springs, elongational flow was not considered since chains do not reach a finite steady state value in their properties when elongational rate exceeds its critical value. Three properties were monitored to follow their rise and decay during the off on off computational experiment: the mean squared end-to-end distance, hr 2 i, the mean squared radius of gyration, hs 2 i, and the birefringence, Dn (operative definitions of such quantities can be found for example in Ref. [9]). Fig. 1a illustrates the response of such a kind of numerical experiments. The relaxation time determined by the foregoing procedure (using the mentioned properties) coincides with the socalled viscoelastic relaxation time, s i (unprimed). That relaxation time can be obtained analytically for an ideal (no-ev) Gaussian chain without HI (Rouse chain) by diagonalization of the Rouse matrix and application of Eq. (1), as explained in the introduction Relaxation times from time correlation function Relaxation times can also be obtained from time correlation functions of the form: q i ðtþ ¼hq i ðt 0 Þq i ðt 0 þ tþi t0 ; ð5þ where q i is the time-dependent normal coordinate vector of the ith Rouse mode. The correlation function involves the scalar product of the value of q i at some given time t 0 by its value when time t has elapsed, averaged over all the possible choices of t 0 along a Brownian trajectory generated in the absence of any perturbation, i.e. at equilibrium. The q i (t) functions decay from the initial value, q i ð0þ ¼hq 2 i i, to q i (t) = 0 for t!1. More particularly, the longest relaxation time of the chain can be computed from the decay of the correlation function of the first Rouse mode, q 1 (t) [,11]. For the specific case of a linear chain, the information on the relaxation time is also contained in the correlation function of the end-to-end vector r: C ee ðtþ ¼hrðt 0 Þ rðt 0 þ tþi t0. There is an exact multiexponential function for C ee in terms of all the relaxation times for Gaussian chains. The longest time constant obtained through this procedure coincides with the so-called longest dielectric relaxation time, s 0 1 (primed). Fig. 2 illustrates an example of the time dependence of the time correlation function of the end-to-end vector plotted in semilog scale. 3. Results Slope~-1/τ 1 ' time Fig. 2. Semilog plot of the time correlation function of the end-to-end vector of a hard Fraenkel spring chain of N = 11 with HI and EV. As above mentioned, when a perturbative external agent is applied to a set of flexible polymer chains, these are oriented, deformed and stretched up to reach the steady-state conformation. Once the perturbation is stopped, properties relax to reach their equilibrium values. This is clearly noticed in the rise and decay of the values of the mean squared radius of gyration observed in Fig. 1(a) (N = 13,

4 5084 J.G. Hernández Cifre et al. / Journal of Non-Crystalline Solids 352 (2006) HI, no-ev, perturbation: shear flow). Property values shown in those figures correspond to averaging over a sample of typically 0 molecules which gave acceptable error bars. The longest relaxation time, s 1, is obtained from a multiexponential fitting of the decay region III (see Fig. 1(a)). The fit was performed using the utilities of the SigmaPlot software. Fig. 1(b) shows in a semilog plot the decay region of Fig. 1(a) shifting the time axis to zero. From that plot it is clearly observed that the case considered fits to a multiexponential (a monoexponential case would result in a straight line). The slope of the tail of that semilog plot is the inverse of the viscoelastic longest relaxation time. On other hand, as explained previously, using a long enough trajectory of a single polymer chain in the absence of external agents, the time correlation function of the first Rouse mode vector (or equivalently for linear chains, the end-to-end vector) is computed and, after multiexponential fitting, the other form of the longest relaxation time, s 0 1,is obtained. Fig. 2 is a semilog plot illustrating the correlation function of the end-to-end vector of a chain of hard Fraenkel springs with N = 11 with HI and EV. As observed, the plot is a straight line which is indicative that the correlation function fits well to a monoexponential. Therefore, the slope of such a plot is readily the inverse of the dielectric longest (indeed unique) relaxation time Effect of polymer topology Table 1 shows the viscoelastic, s 1, and dielectric, s 0 1, longest relaxation times computed for chains with different topology and molecular weight (i.e. number of beads N). In case of a regular star the functionality, F, is the number of arms attached to the core, and the number of beads per arm or arm molecular weight is N arm =(N 1)/F. In addition, we set F = 2 for linear chains and F = 1 for ring (cyclic) chains. All values presented in that table corresponds to ideal (no-ev) chains with FENE (Q max = ) springs including HI. Independently of the topology it is observed that for these chains with approximately Gaussian statistics at equilibrium the relationship s 1 s 0 1 =2 holds within the error. Nevertheless, it must be pointed out that some of Table 1 Viscoelastic and dielectric longest relaxation times for different topologies with HI N F N arm s 1 s 0 1 s 0 1 =s ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0. the relaxation time ratios may depart significantly from the value 2 predicted theoretically for chains with Rouse motion. Recall that results in Table 1 correspond to chains with HI and FENE springs which, if not large enough, may be far from the Rouse model. Larger simulations with larger chains are needed to verified this point. In general terms, as above mentioned, from viscoelastic experiments it is obtained a longest relaxation time half than that obtained from dielectric experiments. As appreciated in Table 1 by comparing relaxation times for stars of different F and linear chains (i.e. stars of F = 2), the value of the longest relaxation time depends on the largest linear piece in the molecules, i.e. the arm molecular weight, N arm, rather than on the global molecular weight N; compare in Table 1 rows corresponding to N arm = 6 (case N = 13 with case N = 25) and rows with N arm = 12 (case N = 25 with case N = 49). Fig. 3 contains log log plots of the longest viscoelastic relaxation time vs. the molecular weight, N, for the different topologies shown in Table 1. As observed, the power law s 1 / N a with a 3/ 2 predicted theoretically for linear chains when HI is present [2] holds quite well regardless the topology. These findings are in agreement with previous simulation results of our group [12] which clearly showed that under elongational flow, the critical gradient, _e c, scales with N arm to the power of 3/2 and that the dimensionless quantity _e c s 1 is independent on N arm and on topology acquiring a value about 0.5. In Table 2 we show the ratio of the dielectric longest relaxation time for linear chains, s 0 1;l, to that of ring chains s 0 1;r with same N. There we find a ratio about 3 which is in agreement with results found in [13] Influence of method to determine relaxation time and connector stiffness Following we show values of both dielectric and viscoelastic longest relaxation time, s 1 and s 0 1, for a linear chain τ F=1 F=2 F=3 F=4 F=6 Slope = 3/2 0.1 Ν 0 Fig. 3. Log log plot of the dependence of the viscoelastic relaxation time on molecular weight for both linear and non-linear FENE (Q max = ) chains with HI and no-ev. Error bars are smaller than symbol size.

5 J.G. Hernández Cifre et al. / Journal of Non-Crystalline Solids 352 (2006) Table 2 Ratio of dielectric longest relaxation time for linear and ring chains of same N N s 0 1;l =s0 1;r ± ± ± ± ± ± 0.2 with N = 11 and with HI. We employed springs with different stiffness as specified in the section of methodology, namely FENE, Hookean and hard Fraenkel springs which follow Eqs. (2) (4), respectively. Besides, different methods to determine the longest relaxation times were evaluated. Results are summarized in Tables 3 5. Tables 3 and 4 show the viscoelastic longest relaxation time, s 1, computed by means of different type of rise and decay experiments (Table 3 contains results without EV (ideal chains) and Table 4 with EV). As appreciated, s 1 is practically independent on the perturbation and property employed to evaluate its value. On other hand the stiffness of the spring influences the value of the viscoelastic relaxation time. Chains with small flexibility and extensibility, which would be well described by the wormlike chain Table 3 s 1 of linear chains with N = 11 and HI, without EV and varying the spring type Spring type Property Perturbation Shear Elongational Electric Hookean hr 2 i 1.54 ± ± 0.06 hs 2 i 1.51 ± ± 0.04 Dn 1.46 ± ± 0.07 FENE (Q max = 2.5) hr 2 i 1.01 ± ± ± 0.05 hs 2 i 1.02 ± ± ± 0.07 Dn 0.90 ± ± ± 0.06 Hard (H = 0) hr 2 i 1.45 ± ± ± 0.05 hs 2 i 1.55 ± ± ± 0.05 Dn 1.47 ± ± ± 0.05 Table 4 s 1 of linear chains with N = 11, HI and EV and varying the spring type Spring type Property Perturbation Shear Elongational Electric Hookean hr 2 i 1.48 ± ± 0.06 hs 2 i 1.44 ± ± 0.07 Dn 1.43 ± ± 0.06 FENE (Q max = 2.5) hr 2 i 0.99 ± ± ± 0.06 hs 2 i 0.96 ± ± ± 0.06 Dn 0.94 ± ± ± 0.07 Hard (H = 0) hr 2 i 1.60 ± ± ± 0.06 hs 2 i 1.54 ± ± ± 0.07 Dn 1.51 ± ± ± 0.08 Table 5 s 0 1 of linear chains with N = 11 and HI and varying the spring type and EV conditions Spring type end-to-end dis. 1st Rouse mode no-ev conditions Hookean 3.2 ± ± 0.2 FENE (Q max = 2.5) 3.1 ± ± 0.3 Hard (H = 0) 3.4 ± ± 0.4 EV conditions Hookean 5.0 ± ± 0.3 FENE (Q max = 2.5) 5.1 ± ± 0.3 Hard (H = 0) 6.6 ± ± 0.4 model (FENE springs with Q max = 2.5), tend to present a smaller value of s 1 than highly flexible chains. Table 5 shows the dielectric longest relaxation time, s 0 1, computed using the time correlation functions of both the end-to-end vector and the first Rouse mode vector, which are, as observed in that table, clearly equivalent for linear chains. The relaxation time computed by this procedure is not much influenced by the choice of the spring potential. Nevertheless EV interactions, i.e. the solvent quality, tend to change the value of the relaxation time. We must consider that dielectric relaxation reflects the relative rotational or orientational motion of the different chain segments. Therefore excluded volume must affect much more the value of s 0 1 than the spring-type does. 4. Discussion Relaxation time values characterizing the dynamics of a polymer chain depend on the kind of property observed. We are able to obtain relaxation times appearing in viscoelastic experiments, s i, by making computational rise-anddecay experiments where polymer chain is perturbed from equilibrium and then let it relax. The nature of the external agent applied to perturb the chain as well as the property evaluated have no significant influence in the value of s i. On other hand, we can get relaxation times coming from dielectric relaxation experiments, s 0 i, by mean of the time correlation function of the first Rouse mode of the chain. Relaxation times are influenced strongly by hydrodynamic interaction which diminishes their value respect to the case in which HI is neglected. HI can be included easily in our Brownian dynamics simulations. Clearly, the chain topology affects the relaxation time values, although the power law s 1 / N 3/2 predicted theoretically for linear chains with HI holds regardless the topology (see Fig. 3). As appreciated in Table 1, the value of longest relaxation time of branched topologies (in this work just regular stars) depends on the largest linear piece in the molecules rather than on the global molecular weight. As observed in Tables 3 and 4, chains with higher connector stiffness present a smaller value of s 1. However, as observed in Table 5, spring stiffness do not influence significantly the value of the dielectric relaxation time, s 0 1.We

6 5086 J.G. Hernández Cifre et al. / Journal of Non-Crystalline Solids 352 (2006) must take into account that the viscoelastic relaxation time measures the stress relaxation so that it is closely connected to the retraction motion of the chain segments (springs) and therefore related to the spring constant H. Thus, it is highly influenced by the type of spring employed in the chain model. On other hand, it seems that EV interactions affect the dielectric relaxation time, s 0 1, whereas they do not influence significantly the value of the viscoelastic relaxation time, s 1. The time s 0 1 is closely related with reorganizations between different parts of the chain and therefore, the impossibility that different chain segments overlap during the rotational relaxation process affects its value. Nevertheless, it must be noticed that chains with N = 11 may be short in order EV effects become important. Therefore, simulations with larger chains are necessary to elucidate the EV influence. In general, differences between s 1 and s 0 1 arise because viscoelastic and dielectric quantities reflect the same global motion in different ways and no general relationship can be established between these quantities. As stated by Watanabe et al. [14], fundamental features of dielectric relaxation are determined by a local correlation function that represents orientational correlation of two chain segments (springs in our model) at two separate times whereas viscoelastic relaxation is determined by a function that represents the orientational anisotropy of each segment at respective time. As a consequence, s 0 1 will be equal or greater than s 1 and the ratio of those quantities is therefore not a constant independent on chain features. Thus, in comparing times of Tables 3 and 4 with the corresponding ones of Table 5, it is relevant to notice that short FENE chains with small Q max and short chains with hard Fraenkel springs, i.e. chains that are most probably far from the Gaussian large chain limit (N!1), do not follow the relationship between viscoelastic and dielectric relaxation time described in the introductory section, say s 1 6¼ s 0 1 =2. Indeed, the relationship s 1 ¼ s 0 1 =2 holds for the limiting case of completely incoherent motion of the chain segments in a short period of time [14], present in the Rouse Zimm dynamics characteristic of Gaussian chains. Acknowledgements This work has been supported by grant BQU from Dirección General de Investigación, MCYT. J.G.H.C. is the recipient of a Ramón y Cajal postdoctoral research contract. R.P. is the recipient of a predoctoral fellowship from MEC. References [1] P.E. Rouse, J. Chem. Phys. 21 (1953) [2] B.H. Zimm, J. Chem. Phys. 24 (1956) 269. [3] H. Yamakawa, J. Chem. Phys. 53 (1970) 436. [4] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, 2nd Ed., Dynamics of Polymeric Liquids, Kinetic Theory, vol. 2, John Wiley, New York, [5] A. Rey, J.J. Freire, J. García de la Torre, Macromolecules 20 (1987) 342. [6] A. Iniesta, J. García de la Torre, J. Chem. Phys. 92 (1990) [7] J. Rotne, S. Prager, J. Chem. Phys. 50 (1969) [8] S. Navarro, M.C. López Martínez, J. García de la Torre, J. Chem. Phys. 3 (1995) [9] J.G. Hernández Cifre, J. García de la Torre, J. Rheol. 43 (1999) 339. [] A. Rey, J.J. Freire, J. García de la Torre, J. Chem. Phys. 90 (1989) [11] A. Rey, J.J. Freire, J. García de la Torre, Macromolecules 23 (1990) [12] J.G. Hernández Cifre, R. Pamies, M.C. López Martínez, J. Gracía de la Torre, Polymer 46 (2005) [13] A. Rey, J.J. Freire, J. García de la Torre, Macromolecules 23 (1990) [14] H. Watanabe, M.-L. Yao, K. Osaki, Macromolecules 29 (1996) 97.