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Eur. Phys. J. B 1, (1999) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Springer-Verlag 1999 A Monte-Carlo study of equilibrium polymers in a shear flow A. Milchev 1,a,

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Eur. Phys. J. B 1, (1999) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Springer-Verlag 1999 A Monte-Carlo study of equilibrium polymers in a shear flow A. Milchev 1,a, J.P. Wittmer,b, and D.P. Landau 3 1 Institute for Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Département de Physique des Matériaux, Université Claude Bernard Lyon I, 696 Villeurbanne Cedex, France 3 Department of Physics and Astronomy, University of Georgia, Athens, Ga. 36, USA Received October 1998 and Received in final form 1 April 1999 Abstract. We use an off-lattice microscopic model for solutions of equilibrium polymers (EP) in a lamellar shear flow generated by means of a self-consistent external field between parallel hard walls. The individual conformations of the chains are found to elongate in flow direction and shrink perpendicular to it while the average polymer length decreases with increasing shear rate. The Molecular Weight Distribution of the chain lengths retains largely its exponential form in dense solutions whereas in dilute solutions it changes from a power-exponential Schwart distribution to a purely exponential one upon an increase of the shear rate. With growing shear rate the system becomes increasingly inhomogeneous so that a characteristic variation of the total monomer density, the diffusion coefficient, and the center-of-mass distribution of polymer chains of different contour length with the velocity of flow is observed. At higher temperature, as the average chain length decreases significantly, the system is shown to undergo an order-disorder transition into a state of nematic liquid crystalline order with an easy direction parallel to the hard walls. The influence of shear flow on this state is briefly examined. PACS Ax Steady shear flows t Polymer reactions and polymeriation 61.5.Hq Macromolecular and polymer solutions; polymer melts; swelling 64.6.Cn Order disorder transformations; statistical mechanics of model systems 1 Introduction Systems in which polymeriation takes place under condition of chemical equilibrium between polymer chains and their respective monomers are termed equilibrium polymers (EP) [1]. The interest to EP from the point of view of both applications and basic research has recently triggered numerous investigation, including computer simulations [4,5] in an effort to avoid difficulties with laboratory experiments [6] and approximations as the Mean Field Approximation (MFA). Recently the basic scaling concepts of polymer physics were tested by extensive Monte-Carlo (MC) simulations of flexible EP on a lattice [4]. The results suggest that despite polydispersity, EP resemble conventional polymers (where the polymeriation reaction has been deliberately terminated) in many aspects. However, dynamic aspects of their behavior may still be very different: for example, the constant process of scission and recombination in EP offers an additional mechanism of stress relaxation [7]. Computer experiments on EP dynamics are already under way [8]. Considerably fewer simulation studies of nonequilibrium properties of EP have been reported [5,9]. Recently observed phenomena such as shear banding a b structure, shear inducing structure and phase transitions [1 15] are not completely understood. An earlier theoretical work [16], for instance, predicted a decrease in average sie of dilute rod-like micelles whereas a later study [17] concluded that rod-like micelles should grow at higher shear rates. Since it is known that viscoelastic surfactant solutions show unusual nonlinear rheology [18], it is clear that much more research in this field is needed before complete understanding of rheological properties of EP is achieved. Since EP behave in many respects as conventional dead polymers [4], comparisons with the latter where much more work on shear flow effects has been done so far, could prove very useful. Thus inhomogeneity of flows, due to the presence of boundaries, and its impact on polymer behavior may be directly observed experimentally by means of evanescent wave-induced fluorescence method [19] that can probe the polymer concentration in the depletion layer adjacent to the walls. Coil stretching of dilute flexible polymers in a flow, diffusion and density profiles as well as slip effects near walls have been treated theoretically [ 3] and by computer simulations [4 6], and as we shall demonstrate below, many of these early results compare favorably with what we observe for EP in the present investigation. 4 The European Physical Journal B In the present study we employ a dynamic Monte- Carlo algorithm in order to study EP properties in shear rate. The flow of the system in a semi-infinite slit of thickness D is induced by applying an external field F with magnitude which changes linearly across the slit and is parallel to the hard walls of the container. Thus the jump rate of the monomers becomes biased along the x-axis and a flow of the system through the periodic boundary sets in. One should emphasie that such an investigation should focus on the linear response in a laminar shear flow. MC methods cannot account for hydrodynamic interactions in principle and the transition from laminar to turbulent flow can be simulated by means of Molecular Dynamics (MD) only. The linear response breaks down at field intensities when the maximum flow velocity is attained, i.e. when all 1% of the random jumps along the x-axis are forced to occur, say, in positive direction. Any further increase of the field F will then fail to accelerate the particles any further. Even with these limitations, however, it appears that this kind of MC simulation of EP in a shear flow is warranted, given the considerably longer time periods or systems sies a MC methods may handle as compared to MD. All Monte-Carlo studies of EP so far have been performed on a cubic lattice either exploiting an analogy of the Potts model of magnetism to random self-avoiding walks on a lattice [1,11], or using the Bond Fluctuation (BFL) Model [4, 9]. These lattice models were developed and extensively used for monodisperse systems of conventional polymers and are known to faithfully reproduce their dynamic (Rouse) behavior. For the purpose of shear flow studies a disadvantage of these models, due to the discrete structure of the lattice, appears obvious: monomers would block each other on the lattice at higher shear rates. Random jumps would have to be of the sie of single monomers only, and, last not least, the artificial cubic symmetry would predetermine ordering effects along the three major axes of the lattice [7] thereby questioning possible phase transitions into liquid crystalline order. In the present work we employ an off-lattice model of EP, designed to overcome these and other shortcomings of previous lattice models and to serve in examining the role of polymers (semi)-flexibility. An off-lattice model should be a better tool in dynamic studies of a broader class of soft condensed matter systems where bifunctionality of the chemical bonds might be extended to polyfunctional bonds, as this is the case in gels and membranes. A comprehensive comparison of this off-lattice algorithm to earlier lattice models [8] shows that all properties of EP derived in former investigations, are faithfully reproduced in the continuum too. Description of the model As in our earlier off-lattice bead-spring model of conventional polymer chain [8,9], a coarse-grained polymer chain consists of l beads or effective monomers. These are connected by springs which represent effective bonds and are described by a FENE (finitely extendible nonlinear elastic) potential: [ U FENE (r) = k ( ) ] r r R log 1 J, (1) R for R r r R, U FENE (r) =, otherwise where r is the distance between two successive beads, r =.7 is the unperturbed bond length with maximal extension l max, R = l max r =.3, and k/ =(inour units of energy k B T =1.) is the elastic constant of the FENE potential which behaves as a harmonic potential for r r R. ThusU FENE (r r ) k (r r ) but diverges logarithmically both for r l max and r l min = r l max. We choose our unit of length such that l max =1 and then the hard core diameter of the beads l min =.4. All lengths as, e.g. the linear sie of the simulational box, are then measured in units of l max. According to equation (1) the net gain of energy of a monomer which forms a bond with a nearest neighbor at distance r is then equal to the bond energy J. In EP these strong attractive bonds between nearest neighbors along the backbone of a chain are constantly subject to scission and recombination. In the present model only bonds, stretched a distance r beyond some threshold value, r break =.8l max, attempt to break so that eventually an energy U FENE (r) in the interval between and J could be released if the bond is broken. Each monomer has two unsaturated bonds which may be either engaged in forming a strong saturated bond between nearest neighbors along the backbone of a chain (when the originally unsaturated bonds of such neighbors meet and become a parallel pair) or remain free (or dangling ) as in the case of chain ends or non-bonded single monomers. In order to create a bond, however, the respective monomers must approach 1 each other within the same interval of distances r break r l max where scissions take place. While covalent bonds are thus constantly broken or created during the simulation, we would like to emphasie that no formation of ring polymers is allowed and this condition has to be observed whenever an act of polymeriation takes place. The non-bonded interaction between monomers is described by a Morse potential, U M (r) = exp [ a (r r min )] exp [ a (r r min )], for r r min , () where r min =.8, and the large value of a =4makes interactions vanish at distances larger than unity, so that an efficient link-cell algorithm [8] for short-range interactions can be implemented. In the present study we maintain our system in the good solvent regime and, therefore, only keep the repulsive branch of equation (), shifting it up the positive y-axis so that U M (r) =for 1 Recombination for r r break would violate detailed balance if scissions occur at r r break only. A. Milchev et al.: A Monte-Carlo study of equilibrium polymers in a shear flow F() new position, r new : W = rnew r old Fdr = x F ((x))dx. (6) Since the potential of the field F, equation (3), is not scalar, W depends on the path followed by the monomer during a jump r old r new. During a jump this path (x) is a straight line, (x) = x x + old, so that from equation (3) one obtains for the work W = x F x dx = B new + old x = F x, (7) x Fig. 1. Definition sketch for external field variation, F x(), causing flow in x-direction between infinite parallel plates. r r min. The radii of the beads and the interactions, equations (1, ), have been chosen such that the chains may not intersect themselves or each other in the course of their movement within the box, so that excludedvolume interactions as well as the topological connectivity of the macromolecules are allowed for. We introduce the shear rate B by defining an external field F whose only component is directed along the x-axis and changes linearly along the -dimension of the box Figure 1: df x () F x () =B( Z max /), = B (3) d so that the bias changes sign at the middle of the box Z max /. A standard Metropolis algorithm governs monomer displacements, whereby an attempted move of a randomly selected particle in a random direction is taken from a uniform distribution within the interval 1 x, y, 1. The presence of impenetrable walls at = and = Z max is observed by rejection of all those jumps of the monomers which would otherwise cause them to leave the box through the planes at the bottom and the top. Thus jumps are attempted with probability P att ( x) = { 1, for 1 x 1, otherwise and accepted with probability, equal to P acc ( x) = exp[ (E new + W E old )/k B T ], for E new + W E old, (4) P acc ( x) = 1 otherwise, (5) where E new and E old are the energies of the new and old system configurations, and W is the work, performed by the field when a monomer jumps from position r old to a where F denotes the average from the values of the field in positions r old and r new.with F, as defined in equation (7), one satisfies the condition of microscopic reversibility with respect to the movements of the particles. Note that it is the microscopic reversibility which requires the use of F, rather than F old, for instance, in the determination of W. This becomes immediately obvious by considering the displacement of a single non-interacting particle in the external field: for a self-consistent algorithm, the update rules for the forward and reverse moves have to be identical (time reversible). The field, F, is thus introduced in the system as an additional term in the Boltmann probability in equation (5) whereby the probability for jumps along or against the field becomes then strongly biased. This term makes the energy in the Boltmann factor in equation (5) a monotonously decreasing function of x. The Metropolis algorithm tends to find the minimum of this energy thus continuously driving the system to an unreached minimum which gives a continuous flow in x direction. One can readily estimate the average jump distance, δx, if W is assumed to be larger than the microscopic interactions U FENE and U M. Setting E new E old = for simplicity in equation (5), with F F one has 1 xpacc (x)dx δx = Pacc (x)dx = x exp( Fx)dx + xdx 1 1 exp( Fx)dx + dx 1 = 1 8 +[1 (1 + F ) exp( F )]/F 1 +[1 exp( F )]/F (8) which yields δx =forf =andδx 1 4 for sufficiently strong fields F, (the sign of δx depending on which half of the box is considered). Thus δx remains bound from above, no matter how strong the applied bias is chosen. In order to provide for the reversibility of scission and recombination events, a Monte-Carlo time step (MCS) has been performed after N particles of the system are randomly chosen and attempted to move at random whereby existing bonds are kept as they are. After that N monomers are again picked at random, i.e. one Of course, since the microscopic interactions are always present, this result remains a rough estimate only whose validity can be checked by the simulation. 44 The European Physical Journal B of the two bonds of each is randomly chosen and, depending on whether this bond exists and points to an existing neighbor along the same chain, or it doesn t, the bond is attempted to break or to create. Thus, within an elementary time step (MCS) N random jumps and as many attempts of bond scission/recombination are carried out, each subject to the Boltmann probability that the respective attempt is successful. It is clear that in a system of EP where scission and recombination of bonds are constantly taking place the particular scheme of bookkeeping is no trivial matter. Since the identity of a particular chain, or monomer affiliation, is in principle preserved for no more than one MCS, the data structure of the chains can only be based on the individual monomers (or, rather, links) as suggested recently [4]. Thus each monomer has two links. The links associated with a given monomer are pointers which may either point to another link or to nowhere. In the latter case a link then represents an unsaturated dangling bond. Thus a large number of particles may be simulated at very modest operational memory. Results in the present study involve systems of up to N = The simulational box (slit) sies are typically D where D is the width of the slit in -dimension. The total density of the monomers c is then defined as the number of monomers per unit volume. During the simulation the whole system is periodically examined, the number of chains with chain length l, square end-to-end distance, R e, gyration radius, R g, center-ofmass coordinates, displacements, etc., are counted and stored. Because of the semi-periodic (in x- and y-direction) boundary conditions the interactions between monomers follow the minimum image convention. The computation of the conformational properties of the chains as R e,forinstance, then implies a restoration of the absolute monomer coordinates from the periodic ones for each repeating unit of the chain. Technical details of this new algorithm will be presented elsewhere, here we will note only that the high efficiency in code performance is achieved by extensive implementation of integer arithmetic in this off-lattice model based largely on binary operations with variables. Thus, for example, the most heavily involved (modulo) operations which provide periodicity of coordinates and the minimum image computation for distances turn out to be redundant. 3 Simulational results 3.1 Velocity profiles As mentioned in Section, we create a shear flow in our system by applying an external field with constant gradient along the -axis of the box so that the flow is oriented along the x-direction cf. equation (5), parallel to the hard walls at =and = D. Thelowerhalfof the box would then flow in positive, the upper one in negative x-direction. It is expected that in the immediate vicinity of the walls the flow might be somewhat distorted due to walls impenetrability. Below, in Figure a we plot the mean jump distance per MCS, δx, measuredalong the x-axis for different values of B. The -coordinates of these successful jumps are taken from the respective - coordinate of the monomers. Evidently, in a wide channel with D = 3 only for sufficiently weak field B.3 the average jump distance grows linearly with respect to the half-width of the box (for B = it is ero). For B .3 distortions in the δx profile set in because the maximal jump distance is limited to.5, as mentioned in Section. For a more narrow slit of width D = 16 the region of linear response would then extend to higher values of F.7. Therefore most of the simulational results in what follows are derived for D = 16. Figure b then demonstrates that the velocity changes linearly across the slit for sufficiently small values of the field B. In the broad channel, D = 3, at =,D for B =.5 one gets F = 8 from equation (3) so that the average jump distance there according to equation (8) should be δx ±.178. The value of δx at the borders of the box, as seen from Figure a, confirms this estimate demonstrating that the role of the microscopic interactions U M and U FENE is small. The presence of the walls is felt in their immediate vicinity and some local distortion of the displacement profile appears increasingly pronounced with growing B although it remains spatially contained in a layer of thickness roughly equal to monomers diameter. It is interesting to note that this small increase of δx (and, therefore, of velocity) immediately at the walls resembles the so called slip effect [3,4] in simple shear flow of dilute polymer solutions in a narrow channel. This slip effect can be explained intuitively by the fact that the polymer molecules near the wall align themselves more strongly with the flow than those away from the wall, and are thus transporting less flow-wise momentum across the flow than would otherwise be the case. Indeed, in our Monte-Carlo model the attempted jumps which would otherwise bring the monomers through and beyond the walls of the slit are always rejected. Since the molecules cannot penetrate the wall, their concentration is reduced at the wall, so that their contribution to the viscosity is further diminished. Using a model of dumbbells in parallel wall shear flow, one can calculate [1,3,4] both the nonlinear velocity profile of the suspended solutions as well as the center of mass concentration profile between the two walls we shall see in the next section that the latter is qualitatively reproduced by our simulational results. 3. Effect of shear rate on average chain length and molecular weight dis

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