Dynamic Properties of Linear and Cyclic Chains in Two Dimensions

Article pubs.acs.org/Macromolecules

Dynamic Properties of Linear and Cyclic Chains in Two Dimensions. Computer Simulation Studies Piotr Polanowski,† Jeremiasz K. Jeszka,‡ and Andrzej Sikorski§,* †

Department of Molecular Physics and ‡Department of Man-Made Fibers, Technical University of Łódź, 90-924 Łódź, Poland Department of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland

§

ABSTRACT: Comparison studies of athermal cyclic and linear polymer chains systems on the two-dimensional triangular lattice are presented. We studied various polymers concentrations with respect to an explicitly considered solvent. The simulation model used is the dynamic lattice liquid model (DLL). This model can work with the highest possible density where all lattice sites are occupied and it allows to take into account coincidences of elementary molecular motion attempts resulting in local cooperative structural transformations. Both static and dynamic properties of the model systems are characterized. We focus on differences between the dynamic behavior of rings and linear chains in solution and how the structure of these objects influence the dynamics of the inert solvent. The simulation results reflect molecular packing and other properties of monomolecular polymer layers, which can be relevant for these, obtained in some thin film formation techniques.

1. INTRODUCTION Polymer rings are interesting and mysterious objects from many points of view: simulation, theoretical and experimental.1−5 This interest implies from a simple fact that ring polymers do not have chains ends. These macromolecules can now be precisely synthesized and characterized.6−9 The structure of cyclic chains and their dynamics properties were the subjects of numerous experimental,10−18 theoretical,19−29 and simulation studies30−62 although the discussion about their properties is not closed yet. The two-dimensional polymer systems containing cyclic macromolecules were also a subject of experiments63−65 and theoretical considerations.66,67 Computer simulations of strictly two-dimensional cyclic macromolecules and of strongly adsorbed three-dimensional rings were not so numerous.39,68−74 They mainly showed that the adsorption of rings is considerably higher than that of linear and starbranched chains. The mobility of rings confined to a thin film was found to slow down during the annealing (or when the strength of the adsorption increases) and they were always slower than linear chains.39,68 The main problem which we would like to discuss in this article is the following: what the dynamic behavior of cyclic chains in solutions looks like and how it is related to the corresponding solution which contains linear chains in the twodimensional case. The cyclic chains we have taken into consideration are “collapsed” in the sense that they did not contain any solvent molecule inside the ring (thus, the area within the chain’s contour was zero−see Figure 1). The studies of such rings were suggested by the fact that they are intermediate objects between coiled linear flexible chains and rigid rods similar to needles.75 Moreover, our studies are motivated by the fact that in the two-dimensional case segments of polymer chains create impenetrable barrier for © XXXX American Chemical Society

solution molecules. In the ring case this should provide stronger changes of solvent dynamics contrary to the threedimensional case where solvent molecules can penetrate surface created by the contour of cyclic chain. As one can see in the two-dimensional case a solvent molecule cannot be arrested in the inner area of a polymer ring as in the case of twodimensional linear chains. Thus, the possibility of participating of solvent molecules in cooperative motions of polymer segments is strongly limited. This difference between linear and cyclic chains is clearly visible in Figure 1. Another problem is connected with the fact that small rings are not so compressible and it leads to the interesting dynamical behavior of both solvent molecules and polymer segments. Moreover, the above task partially corresponds to a well-known problem of penetration of small solvent molecules in polymer matrices. This problem was regarded at first using computer simulation by Mü l ler-Plathe76 where the hopping as a transport mechanism was used. The results in present work were obtained using another transport mechanism based on cooperative motion. In this paper, we present the results of the simulations based on the dynamic lattice liquid model (DLL) model.77−79 Thus, the model in a natural way allows to investigate polymer chains with a surrounding medium. Contrary to other models and calculation methods of soft matter systems at high densities the DLL allows to take under consideration the correlation in motion of all elements of the system for quite long times and large systems.80 Our study is focused on the self-diffusion of linear chains and rings as a function of polymer concentration Received: March 4, 2014 Revised: May 28, 2014

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dx.doi.org/10.1021/ma500460b | Macromolecules XXXX, XXX, XXX−XXX

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required. Thus, we had to apply the DLL algorithm, which is more general; i.e., it works in the same manner regardless of the concentration. However, it is not so fast on a sequential machine. The DLL model is based on a lattice structure with beads representing small solvent molecules or polymer elements (segments). Positions of beads are regarded as coinciding with lattice sites. The assumption of dense packing of molecules leads to the consideration of a system with all lattice sites occupied by polymer segments or by the solvent. It is also assumed that the system has some excess volume, so that each molecule has enough space to vibrate around its position defined by the lattice site only. The molecules, however, cannot move easily over a longer distance, because all neighboring lattice sites are occupied. Nevertheless, long-range mobility can take place in such a system. The DLL model answers the following question: under which conditions are molecular translations over distances exceeding the vibrational range possible? Each, large enough, displacement of a molecule from its mean position is considered as an attempt of movement to a neighboring lattice site. For simplicity, directions of the move attempts are assumed only along the coordination lines, but they are independent and randomly distributed among q directions, where q is the lattice coordination number. Only those attempts can be successful which coincide in such a way that, along a path including more than two molecules, the sum of displacements is close to zero (the condition of continuity). This results in displacements of beads along self-avoiding closed paths. The model described above has been implemented as a dynamic Monte Carlo simulation algorithm for polymers in a solvent confined to a two-dimensional layer. The system is built on a triangular lattice. The beads are regarded as solvent molecules or segments of polymer chains and they occupy all lattice sites. Beads forming polymers are connected by nonbreakable bonds and constitute linear or cyclic polymers. Motion attempts are represented by a field of randomly chosen unit vectors, which are assigned to beads and point directions of attempted displacements. An example of such assignment of attempted directions of motion is shown in Figure 2 for a system representing polymer solution on a triangular lattice. All beads which do not contribute to correlated sequences (circuits) satisfying the continuity condition are nonmoveable at the moment. This occurs in the cases 1−4 illustrated in Figure 2. After setting to zero all vectors giving unsuccessful attempts, only vectors contributing to a number of closed loops remain. They constitute traces for possible rearrangements (the case 5 in Figure 2). If an athermal system is considered, all possible rearrangements are performed by shifting beads along the closed loop traces, each bead to a neighboring lattice site. Thus, the scheme of this algorithm is the following: (i) the generation of the vector field representing attempts of movement, (ii) elimination of nonsuccessful attempts, and (iii) replacing beads within closed loop paths. A single Monte Carlo step consists of the above actions (i−iii). A molecular or macromolecular system treated in this way can be regarded as provided with the dynamics consisting of local vibrations and occasional diffusion steps resulting from coincidence of attempts of the neighboring elements to displace beyond the occupied positions. Within a longer time interval, this kind of dynamics leads to displacements of individual beads and chains along random walk trajectories with steps distributed randomly in time. A discussion concerning the detailed balance and the ergodicity in this model is presented in ref 80. It should be stressed that the algorithm presented above

Figure 1. Illustration showing systems with linear chains (upper) and rings (lower). Both architectures concern the chain length N = 32 and the polymer concentration φ = 0.5.

in “‘ideal’” polymer solutions where the behavior is controlled by the strongest interactions, i.e., the excluded volume, the dense packing and the nonbreakable bonding of segments forming chains. There are no other specific interactions between polymer segments and the solvent. Simulations are performed in two dimensions, i.e., for the case, which is considered as more strongly influenced by mutual interactions between polymers and the solvent. We took under consideration rather short chains where the influence of the macromolecular topology should be essential. The studies of longer chains within the frame of the DLL model are still beyond the computational capabilities.

2. SIMULATION METHOD There are two Monte Carlo simulation algorithms on a lattice that can be used for studying dense polymer systems where the entire space is filled by objects (polymers segments and solvent molecules). The first one is the cooperative motion algorithm (CMA), which employs the cooperative rearrangements along closed trajectories.80 The second algorithm is the DLL in which the cooperative motion is organized along loops of different lengths.77−80 CMA is considerably faster than DLL, which enables studies of longer chains. CMA is a sequential algorithm with detailed balance and microscopic reversibility, which properly reflects the dynamics of the system containing only polymer chains (the concentration equal 1.0). On the other hand, the dynamics in system, which beside polymers contains solvent molecules arises some doubts and in order to overcome these difficulties an additional assumption in the model is B


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⟨R g 2⟩ =

1 N

N

∑ ( ri ⃗ − rcm⃗ )2 i=1

(3)

where rc⃗ m is a coordinate of the chain center of mass (it was determined for both chain architectures under consideration). In principle, linear chains scale with the length as ⟨Rg2⟩ ∝ ⟨Ree2⟩ ∝ (N−1)2v, but rings as ⟨Rg2⟩ ∝ ⟨Rnn2⟩ ∝ N2v because in the latter case the number of bonds is equal to the number of beads. Figure 3a presents log−log plots of ⟨Rg2⟩ for linear chains and rings, ⟨Ree2⟩ for linear chains and ⟨Rnn2⟩ for rings versus the

Figure 2. Illustration of the vector field representing attempts of molecular displacements toward neighboring lattice sites in the DLL model. Labeled areas represent various local situations: (1) elements try to move in the opposite direction (unsuccessful attempt), (2) an attempt of motion starts from an element that when moved would not be replaced by any of its neighbors (unsuccessful attempt), (3) attempted movement would lead to a break of bonds in the polymer chain (unsuccessful attempt), (4) the solvent particle would jump through a polymer bond (unsuccessful attempt), and (5) each element replaces one of its neighbors (successful attempts).

is strictly parallel; i.e., all steps are performed simultaneously with regard to all system elements. Figure 3. (a) Chain length dependencies of the mean square end-toend distance ⟨Ree2⟩ for linear chains, ⟨Rnn2⟩ the mean square diameter for rings and the mean square radius of gyration ⟨Rg2⟩ for both architectures of chains. Dashed lines indicate asymptotic slopes 1.5 and 1.0, which are characteristic for dilute and dense systems in two dimensions for linear chains, respectively. The symbols are explained in the inset. (b) Scaling exponent 2ν as a function of the polymer concentration for chains and rings. The errors were not marked, as they are smaller than the size of symbols.

3. RESULTS AND DISCUSSION The simulations were performed for systems containing a number of chains of identical length and shape. We studied systems of linear and cyclic chains consisting of N = 8, 16, and 32 beads. The DLL model, although powerful and useful, appeared to be too slow for studying the long-time dynamics of dense systems containing longer macromolecules. The size of the simulation box was L× L where L = 256; i.e., it was considerably larger than the size of chains under considerations. The density of the polymer in the system φ was defined as the ratio of the sites occupied by the polymer beads to the total number of lattice sites in the simulation box, and thus, φ = nN/ L2, where n is the number of chains (each of identical length) in the system. If the volume fraction of the macromolecules in the solution is smaller than unity, lattice sites unoccupied by polymer segments are filled with single beads representing solvent molecules. Thus, the concentration of solvent in the system was φs = 1 − φ. In this way the system can always be regarded as a dense in which all lattice sites are occupied by either the suspension particles or the solvent particles. 3.1. Static Properties. The conformational state of chains in the simulated systems is characterized here by the mean square values of the end-to-end distance for linear chains ⟨Ree 2⟩ = ⟨( r1⃗ − rN⃗ )2 ⟩

chain length N − 1 for the entire range of the polymer concentration, i.e. between φ = 0.05 and 0.95. One can observe that the size of cyclic chains is always considerably smaller than that of linear ones (we treat the mean-square radius of gyration as a measure of size). The increase of the polymer concentration leads to the contraction of chains but the influence of the polymer concentration on chain size is very weak in the case of rings. However, in the case of the linear chain lengths we studied, i.e., the short ones only, it is difficult to determine their scaling behavior precisely. One can observe that linear chains reproduce the scaling behavior ⟨Rg2⟩ ∼ (N − 1)2v quite well and the scaling exponent 2ν is located between 1.5 and 1 expected in this case. Figure 3b shows the changes of the exponent 2ν as a function of polymer concentration φ. The values of the exponent determined for linear chains change from 1.42 ± 0.01 (for the dilute system) to 1.11 ± 0.01 (for the dense system). However, one has to remember that our chains are rather short and this is apparently the reason why the exponent does not approach the exact values 1.5 and 1. Parts a and b of Figure 3 show also that the cyclic chains scale with N with the considerably lower exponent than linear chains do. It was expected because they are similar to lattice animals, which scale according to the theory and computer simulations with the exponent 2v ≈ 1.25 and 1.28 respectively for single

(1)

and the mean square diameter, i.e., the distance between the first and (N/2)th element of rings ⟨R nn 2⟩ = ⟨( r1⃗ − rN⃗ /2)2 ⟩

(2)

where r1⃗ and rN⃗ are space coordinates of chain ends. The next parameter considered was the mean square radius of gyration ⟨Rg2⟩ C


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objects.81,82 The exponents 2ν for rings obtained in our simulations vary from 1.25 ± 0.01 (for the dilute system) to 1.17 ± 0.01 (for the dense system). The character of changes of the exponent 2ν with the polymer concentration is different for both objects studied: it rapidly decreases for linear chains while for the rings this decrease is observed for the concentrated solutions only and it is no so rapid which is confirmed by results described below. The above results should be interpreted with caution because in our case we are dealing with short chains while the scaling theory gives good results for N → ∞. Recent computer simulations of a similar model founded this exponent located between 1.30 and 1.04 for similar model of rings of longer (consisting of up to N = 512 segments) at the polymer concentration φ = 0.05 and 1.0 respectively.75 Discontinuous molecular dynamics of twodimensional rings with solvent molecules (at low solvent concentration φ = 0.16) showed that the exponent 2ν changes from 2.0148 (the maximum possible number of solvent molecules inside the chain contour) to 1.330 (no solvent molecules inside the chain contour);72 the latter value is quite close to our findings. It is worth mentioning that experiments on single circular DNA adsorbed on mica also founded the exponent 2v = 1.5.64 Figure 4 depicts the size of chains characterized by ⟨Rg2⟩ as functions of the polymer concentration. One can observe that

The behavior of the static form factor (sff) is a good illustration of changes the shape and the size of chains as a function of the polymer concentration. This parameter is usually defined as S(q ⃗ ) =

1 N

N

∑

⟨exp(iq ⃗( rn⃗ − rm⃗ )⟩

n.m=1

(4)

where q⃗ is the scattering vector. The form factor values determined for the simulated systems containing chains N = 32 as a function of qRg are presented in Figure 5a. The plots of the

Figure 5. (a) Static form factor of linear chains (open symbols) and rings (solid symbols) for the chain length N = 32 at various polymer concentrations (φ = 0.1, 0.5, and 0.95) as a function of qRg. Red solid lines represent the Gaussian chain (Debay function) and dashed line indicates the asymptotic slopes −2. (b) Plot of the static form factor in the Kratky representation, with values multiplied by the factor (1 − φ) for better visualization and with red solid lines represents Gaussian chains.

sff parameter are in good agreement with the results obtained for ⟨Rg2⟩ where the changes in the ring structure with increase of the polymer concentration are weaker than in the linear chain case. For rings all curves, which represent sff (at concentrations 0.10, 0.50, and 0.95) are very close while systematic changes with the increase of the concentration are observed for linear chains. In other words in the latter case the slope of the sff parameter shifts toward the direction of higher values of q and becomes steeper which means that the structure is more compact. Red solid lines in parts a and b of Figure 5 show the behavior of Gaussian chains. The static form factor of Gaussian chains is represented by the Debye function obtained as84

Figure 4. Polymer concentration dependencies of the mean square radius of gyration ⟨Rg2⟩ for both architectures of chains. Rings are marked by solid symbols and linear chains by open symbols. The dashed line indicates the slope −1 from the de Gennes scaling theory.

for all chains lengths and polymer concentrations linear chains are more expanded than the rings. The contraction of the chain size induced by the increased polymer concentration is higher for the systems containing linear polymers compared to rings. The longer is the polymer the higher is the contraction. Moreover, for longer linear chains (N = 16 and N = 32) one can distinguish three regimes of different behavior in the entire concentration range. The first one (I) can be regarded as the dilute regime, in which the polymer concentration does not influence the chain dimension. The second one (II), where the reduction of chain size begins can be treated as the semidilute regime. In the third one (III), which corresponds to the concentrated solution, the radius of gyration scale with polymer chain concentration like φ−1 which is consistent with the prediction of de Gennes.83 One can observe that the influence of the polymer concentration on the size of rings in the regarded range of chain length is quite different. Collapsed rings are considerably less compressible with the increasing polymer concentration, which is consistent with the results presented in Figure 3b.

S De(q ⃗) =

2(N − 1) (exp( −q2⟨R g 2⟩) − 1 + q2⟨R g 2⟩) (q2⟨R g 2⟩)2 (5)

At low scattering angles values of S(q⃗) obtained from the simulations for both linear chains and rings reproduced the Guinier regime S(q) ≈ exp(−(q2⟨Rg2⟩)/3) which is visible on the Kratky plot in Figure 5b. At high scattering vectors the values of S(q⃗) obtained from the simulations for linear chains do not follow q−2 dependence: the values of S(q⃗) for the dilute (φ = 0.05) and the semidilute (φ = 0.50) concentrations become higher and do not reach the plateau as the Gaussian D


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chains do. This means that the simulated linear chains are more rigid than Gaussian ones, which is visible in the Kratky representation presented in Figure 5b. For the concentration φ = 0.95 we obtained the behavior of S(q)⃗ similar to that for the Gaussian chain and this result is in good agreement with results obtained in a beautiful experiment performed by Li et al.85 In the case of rings one can observe in Figure 5a that S(q⃗) is slightly smaller than that of the Gaussian chain in the entire range of scattering vectors. It means that rings at all concentrations are more collapsed and more similar to disks than Gaussian chains. This picture is also visible for qRg < 1 in the Kratky plot in Figure 5b. The above observations can be confirmed by the investigation of rings and chains asphericity, which is discussed below. Theoretical predictions for the form factor of cyclic chains were given many years ago by Casassa, which can be written in the following normalized form:23,28 S(q) =

1 exp( −t 2) t

∫0

t

exp(x 2) dx

A2 =

⟨(λ 2 − λ1)2 ⟩ ⟨(λ 2 + λ1)2 ⟩

(9)

This parameter takes value A2 = 1 for a fully extended chain and A2 = 0 for a disk. Figure 7 presents log−log plots of the asphericity of linear chains and rings as a function of the polymer concentration.

(6)

where t = (q⟨Rg⟩)/2 but one has to remember that these results are valid near the theta temperature only. Figure 6 depicts the

Figure 7. Polymer concentration dependencies of the asphericity for both architectures of chains. Rings are marked by solid symbols and linear chains by open symbols.

Rudnick and Gaspari in86 found that for the infinite linear chain in the case of no excluded volume eq 9 reduced to A2 = (2(d + 2))/(5d + 4) which gives in two-dimensions A2 ≈ 0.57. In the case when the excluded volume is introduced as in our simulations this value can only be higher. One can observe that for linear chain lengths N = 16 and 32 our results are close to predictions of Rudnick and Gaspari.86 The plots in Figure 6 indicate also that in all cases the shape of rings is more similar to disk than the shape of linear chains; i.e., the collapsed twodimensional rings behave similarly to the three-dimensional regular rings.42,57 The higher is the polymer concentration the more spherical is the chain and the influence of the chain length on the asphericity is definitely stronger for rings. The above static properties obtained in our simulations can be compared with the results obtained by other simulation techniques for two-dimensional rings. Molecular dynamics33 and Brownian dynamics31 simulations for short chains give considerably anisotropic chains (A2 changes between 0.65 and 0.63 for chains between N = 8 and N = 32). These techniques were able to show a slight decrease of anisotropy with the polymer concentration, as they are applicable up to semidilute solutions only. Theoretical exact results for short nonexcluded volume rings are considerably lower:85 one can interpolate A2 = 0.61 and 0.58 for N = 8 and 32, respectively. Snapshots of the regarded systems depicted in Figure 8 confirm this behavior and shows that the size of collapsed rings is not sensitive on the polymer concentration. 3.2. Dynamic Properties. The influence of the polymer concentration and the chain length on dynamic properties in the two-dimensional solutions can be characterized by the following quantities (i) The mean square displacements of polymer segments ⟨ΔRmon2⟩, the mean squared displacements of the centerof-mass ⟨ΔRcm2⟩of chains, and the mean squared displacements of solvent molecule ⟨ΔRsolv2⟩ defined as

Figure 6. Comparison of the form factor predicted by Casassa for chains and rings (N = 32)28 with the simulation results for the polymer concentration φ = 0.95.

comparison of the normalized values of S(q) for rings and chains shown in Figure 5 with the Casassa form factor computed for values of the radius of gyration taken from simulation for the concentration φ = 0.95. One can observe that Casassa prediction differs from the results for rings what can be expected because our rings are rather specific objects. In order to investigate the geometry of chains and rings more accurately we have also determined the gyration tensor T Tkl =

1 N

N

∑ (rik − rcm,k)(ril − rcm, l) (7)

i=1

where k and l are the coordinates x and y, rik is the kth coordinates of the position ri⃗ and rcm,k is the kth coordinate of the chain center-of-mass. The tensor T has two eigenvalues, which are denoted λ1 and λ2 with the convention λ1 ≥ λ2 and which fulfill the relation R g 2 = λ1 + λ 2

The asphericity A2 was calculated as

(8) 86

E


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corresponding object in model (segments, linear chains or rings). At first we will discuss the differences in dynamical properties of linear chains and rings basing on the exemplary behavior of chains N = 8. It is an interesting case because it shows how the stiffness arisen from topological constraints in collapsed ring and how the absence of ends in cyclic polymers influence the chain dynamics. Figure 9a shows the mean square displace-

Figure 8. Snapshots of systems consisted of rings N = 32 at the polymer concentration φ = 0.1, 0.5, and 0.95 (concentration changes from the left side to the right side, upper panel shows the chains lower shows the rings).

1 nN

⟨ΔR mon 2(t )⟩ =

n

Figure 9. Mean square displacement of the chain’s center of mass (a) and the mean square displacement of solvent molecules (b) for the case of the chain length N = 8. Systems containing rings are marked by solid symbols and those with linear chains by open symbols.

N

2 ⃗ , ij(t ) − rmon ⃗ , ij(0)] ∑ ∑ [ rmon

ments of the center of mass of rings and linear chains for various concentrations of polymer (φ = 0.1, 0.5 and 0.95). One can observe that in all presented cases rings center of mass moves considerably slower than that of linear chains and this difference in the mobility increases with the polymer concentration. Moreover, for φ = 0.95 one can distinguish three different regimes of self-diffusion: short time diffusion, slowing down diffusion and long time diffusion which are very well visible for rings and practically invisible for linear chains. Results observed in Figure 9a reflect the changes in the global and local dynamics of polymer chains with the changes of the polymer concentration. In the case of linear chains in both cases; i.e., at a low and a high polymer concentration, we did not observed drastic differences in the short and long time dynamics. This suggests that the dynamics of linear chains is determined by local dynamics however not too big deviations are observed at φ = 0.95. In case of rings at low concentrations of polymer φ = 0.1 and 0.5, the dynamics of macromolecules is determined by the local dynamic too. For φ = 0.95, one can observe a very strong effect of diffusion slowing down connected with caging of ring polymer by other rings which is a consequence of the stiffens and the absence of free ends. The dynamic behavior of linear chains influences the dynamics of solvent molecules. Figure 9b shows the mean square displacements of a solvent molecule as a function of time in both regarded cases for the same polymer concentrations as in Figure 9a. The mobility of these small molecules is higher more than 2 orders of magnitude when compared with macromolecules. One can observe that for a low concentration of polymer, the dynamics of a solvent particle is practically the same for both cases suggesting that the mobility of solvent in this case is determined by the local dynamics. For φ = 0.95, the slowing down diffusion connected with caging of solvent molecules between polymer chains in both cases is observed,

i=1 j=1

(10)

1 n

⟨ΔRcm 2(t )⟩ =

n

∑ [ rcm⃗ ,i(t ) − rcm⃗ ,i(0)]2 i=1

1 nsolv

⟨ΔR solv 2(t )⟩ =

(11)

nsolv

2 ⃗ , i(t ) − rsolv ⃗ , i(0)] ∑ [ rsolv i=1

(12)

Here rm⃗ on(t), rc⃗ m(t), and rs⃗ olv(t) are vectors representing the mth segment, the chain center of mass, and a solvent molecule position at time t. The above quantities allow us to determine the appropriate diffusion constants. (ii) The autocorrelation function of polymer segments ρb (t ) =

1 nN

n

N

∑ ∑ bj⃗ (t )bj⃗ (0) i=1 j=1

(13)

where bj⃗ is a unit vector representing jth segment orientation. (iii) The autocorrelation function of the end-to-end vector in linear chains 1 ρR (t ) = n

n

∑ R⃗ee(t )R⃗ee(0) i=1

(14)

and the autocorrelation function of the diameter vector R⃗ nm in rings ρR (t ) = nn

1 n

n

∑ R⃗ nn(t )R⃗ nn(0) i=1

(15)

The autocorrelation functions of segments, R⃗ ee and R⃗ nn vectors enable us to determine relaxation times of the F


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where ΔR2 is the mean square displacement of the regarded object in time and D is the self-diffusion constant. The self-diffusion constants of the center of mass for linear chains and rings as a function of the polymer concentration φ for a given chain length N are presented in Figure 11.

however this effect is considerably weaker in linear chains case. Moreover, the short time diffusion of linear chains is substantially higher than that of rings. Further information concerning the dynamics of the system studied can be extracted from the analysis of autocorrelation functions defined by eq 13−15. Figure 10 shows the

Figure 11. Polymer concentration dependencies of the long time selfdiffusion constant (see text for details) for linear chains (open symbols) and rings (solid symbols) presented for various chain lengths.

Figure 10. Segment vector autocorrelation function for N = 8 (a) and Ree and Rnn autocorrelation functions (b). Rings are marked by solid symbols and linear chains by open symbols.

The dependencies of the self-diffusion constant of linear chain and ring on polymer concentration have a similar character for different chain lengths. We can distinguish three regimes of the diffusion constant behavior: the dilute from 0 to 0.3, the semidilute from 0.4 to 0.6 and the concentrated one, which begins at 0.7. In each of these regimes, characteristic dependencies on concentration are observed. However, rings move considerably slower than the corresponding (consisting of the same number of segments) linear chains and the polymer mobility decreases with the increase of the chain length for both species. The transition from the semidilute to the dense regime has a drastic character and takes place at much higher concentration than the percolation threshold. This threshold in the case of single beads without the presence of chains corresponds to the concentration φ = 0.5,81 but for systems containing similar model chains, it varies: it is located between 0.516 (for N = 8) and 0.501 (for N = 32) for rings and between 0.462 (for N = 8) and 0.340 for (for N = 32) for linear chains.33,75 Therefore, it is difficult to connect these changes in the diffusion with percolation processes. It should be rather explained by the sharp limitation of cooperative movement, which is connected with the small number of solvent molecules. Moreover, one can observe “abnormal” slow diffusion for rings with the length N = 8 for high dense system. On the basis of similarities in the concentration dependencies of polymer diffusion constants for various chain lengths, a so-called empirical scaling possibility has been suggested.87 Let us assume that the diffusion constant of the center of mass for an isolated (single) polymer chain depends on the chain length as follows:

autocorrelation function of polymer segments as well as of R⃗ ee and R⃗ nn vectors. One can observe that the relaxation time of segments (Figure 10a), which gives the information about the local dynamics of polymer segments, is considerably different for linear chains and rings: in general, relaxation times of segments in linear chains are shorter; i.e., the relaxation in the latter case is slower. The relaxation time of rings at polymer concentrations φ = 0.1 and 0.5 is practically the same as for linear polymers at the concentration φ = 0.95. The relaxation time of rings at φ = 0.95 is considerably longer than that of linear chains. This observation can be simply explained by the fact that in the case of rings the cooperative movement of its segments is strongly connected with the movement of other elements of chains, whereas in the case of linear chains polymer segments can move cooperatively with solvent molecules easier than in the ring case. The presence of free ends in linear chains is the next reason for this difference in the relaxation times. The relaxation times of the entire macromolecules are represented by autocorrelation functions of R⃗ ee and R⃗ nn vectors, which is depicted in Figure 10b. In this case the time autocorrelation function looks similar for both chain architectures at low concentrations, but at the concentration φ = 0.95, we observed again a big difference in the relaxation times of rings and linear chains. More general treatment of transport properties of considered objects; i.e., linear chains, rings, and solvent require the analysis of self-diffusion coefficients in the whole regarded range of chain length and polymer concentration which were obtained using the Einstein relation ΔR2(t ) = 4Dt , t → ∞

Disol (N ) ∝ g (N )

(17)

Here g(N) is an unknown decreasing function of the chain length. For a polymer solution, we can define a similar diffusion constant dependence but it has to be modified by the factor B(Y), which allows us to take into consideration both the chain

(16) G


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length and the polymer concentration influence on the chain mobility. Thus, we have Dcm(N , φ) ∝

g (N ) B (Y )

(18)

where Y is a scaling variable defined as follows: Y = ϕ (2(N− 1)/N)1/2. Taking under consideration eq 17 and 18, one can write

Disol (N ) ∝ B (Y ) Dcm(N , φ)

(19)

The dashed curve in Figure 12 is a fit function of the form Disol (N ) ∝ 1/[0.59( 2 − Y )]1.9 Dcm(N , φ)

Figure 13. Polymer concentration dependencies of the monomer selfdiffusion constant for linear chains (open symbols) and rings (solid symbols) presented for various chain lengths.

(20)

constant of the center of mass. However, the changes of the monomer mobility with the chain length especially for the dilute case for rings are smaller which is connected with the fact that in the ring case monomers have to take place in cooperative movement of their own ring and their cooperative movement correlated with segments from other rings or solvent molecules is strongly limited. The cooperative movement involving different chains has a crucial importance for fast single monomer mobility of linear chains. The snapshots of the considered systems presented in Figure 9 confirm this difference in mutual orientation and location in the systems of linear and rings chains. Taking into account the results shown in Figures 9b (and similar results obtained for chain lengths N = 16 and 32 which are not presented here) one can distinguish two dynamic regimes for the diffusion of solvent particles for which the mean square displacement increases linearly with the time at short time and at long time (for diluted solution, as shown for the concentration φ = 0.05 in Figure 9b, one regime only can be distinguished). It allows us to determine the diffusion coefficient in the short and long time respectively DST and DLT. The short and long time diffusion constants of a solvent molecule as a function of the polymer concentration for various chain lengths are shown in Figures 14a,b. One can observe that the local dynamics of solvent molecules weakly depends on the

Figure 12. Fit of the empirical scaling relation according to eq 19 to the self-diffusion constants of the polymer center-of-mass as a function of both the concentration and the chain length. The dashed curve is a fit to the simulation results. In the inset the crossover between slow and fast variation of the diffusion constant φc as a function of the chain length N is presented.

From the shape of the function B(Y) in Figure 11, one can see dramatic changes of the centers of mass diffusion constant (for both chains and rings) appear for approximately the same value of Y, independently of the chain length. Considering the dependence of Y on N, we can determine the dependence of the composition φc corresponding to the crossover between slow and fast variation of the diffusion constant on the chain length. The changes of this crossover φc with the chain length N are shown in the inset in Figure 12. It must be pointed, however, that similarly to the static properties discussed above we cannot extend these results to longer chains. Figure 13 shows the polymer segment self-diffusion coefficient calculated from the mean square displacement for linear chains and “collapsed” rings as a function of the polymer concentration. This coefficient was determined from the initial linear part of the mean square displacement plots according to eq 16. The self-diffusion coefficient of a segment for rings is considerably lower than that for linear chains. The character of the changes of the segment self-diffusion constant is similar in both architectures for linear and ring chain similarly to the behavior of concentration dependencies of the self-diffusion

Figure 14. Polymer concentration dependencies of the short (a) and the long time (b) self-diffusion constant of solvent molecules for linear chains (open symbols) and ring (solid symbols) presented for various chain length. H


Macromolecules

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studied. The diffusion of linear chains and rings is mostly Fickian but rings at higher densities behave in a different way: a slowing down was observed for intermediate times. It was shown that this slowing down is not connected with the percolation. The reason for this behavior was apparently screening and caging by other chains. Another interesting piece of information concerned the motion of solvent molecules in the considered polymer systems, where a similar slowing down of the motion was observed at higher ring concentrations. The additional constraint introduced into the model cyclic chains; i.e., the restriction of the phase space to “collapsed” structures only was responsible for these differences too. Therefore, this study confirms that “collapsed” ring chains are efficient barrier for the motion of solvent molecules.

chain length for a given polymer concentration and systematically decreases with the increase of the polymer concentration. Moreover, this dependency is practically independent of the architecture of the chain what is interesting if one remember that linear chains and rings exhibit considerable differences between segment mobility. It confirms that the movement of a polymer segment in this case does not influence the mobility of solvent. In this situation, in both architectures, linear chains and rings polymer segments play only a role of the barrier for the cooperative movement of solvent molecules. This is a consequence of differences between the relaxation time of polymer segments and the time required for the change of a position by a solvent molecule in the result of the cooperative movement, which is much shorter. The behavior of the long time diffusion observed in Figure 14b is much more complicated. In the ring case, the mobility of solvent molecules decreases with the increase of the polymer concentration but practically independently on the chain length. This behavior is similar to the results obtained for local dynamics and described above. However, the long time mobility of solvent in this case is smaller than the local mobility which is connected with the fact that now we observe the whole system where dynamics is mainly determined by global movement of rings. The similar situation can be observed for linear chains but in this case one can find that the mobility of solvent changes with the polymer length for a given concentration. This effect can be expected, as it is a consequence of the fact that the penetration of inner area of rings by solvent molecules is forbidden.

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AUTHOR INFORMATION

Corresponding Author

*(A.S.) E-mail: [email protected] phone: +48 22 822 0211 fax: +48 22 822 5996. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Semlyen, J. A. In Cyclic Polymers, 2nd ed.; Semlyen, J. A., Ed.; Kluwer Academic: Dordrecht, The Netherlands, 2000. (2) Benmouna, M.; Maschke, U. In Cyclic Polymers;,2nd ed.; Semlyen, J. A., Ed.;Kluwer Academic: Dordrecht, The Netherlands, 2000. (3) McLeish, T. Phys. Today 2008, 61, 40. (4) McLeish, T. Science 2002, 295, 2005. (5) Beckham, H. W. Complex Macromolecular Architectures. In Synthesis, Characterization and Self-Assembly; Hadjichristidis, N.,Hirao, A.,Tezuka, Y.,Du Prez, F., Eds.; Wiley: Singapore, 2011. (6) Endo, K. Adv. Polym. Sci. 2008, 217, 121. (7) Kricheldorf, H. R. J. Polym. Sci. Part A 2010, 48, 251. (8) Yamamoto, T.; Tezuka, Y. Complex Macromolecular Architectures. In Synthesis, Characterization and Self-Assembly; Hadjichristidis, N.,Hirao, A.,Tezuka, Y.,Du Prez, F., Eds.; Wiley: Singapore, 2011. (9) Yamamoto, T. Polym. J. 2013, 45, 711. (10) Bielawski, C. W.; Benitez, D.; Grubbs, R. H. Science 2002, 297, 2041. (11) Arrighi, V.; Gagliardi, S.; Dagger, A. C.; Semlyen, J. A.; Higgins, J. S.; Shenton, M. J. Macromolecules 2004, 37, 8057. (12) Gagliardi, S.; Arrighi, V.; Ferguson, R.; Dagger, A. C.; Semlyen, J. A.; Higgins, J. S. J. Chem. Phys. 2005, 122, 064904. (13) Takano, A.; Kushida, Y.; Ohta, Y.; Masuoka, K.; Matsushita, Y. Polymer 2009, 50, 1300−1303. (14) Habuchi, S.; Satoh, N.; Yamamoto, T.; Tezuka, Y.; Vacha, M. Angew. Chem.,Int. Ed. 2010, 49, 1418. (15) Robertson, R. M.; Smith, D. E. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 4824. (16) Robertson, R. M.; Smith, D. E. Macromolecules 2007, 40, 3373. (17) Takano, A.; Ohta, Y.; Masuoka, K.; Matsubara, K.; Nakano, T.; Hieno, A.; Itakura, M.; Takahashi, K.; Kinugasa, S.; Kawaguchi, D.; Takahashi, Y.; Matsuhita, Y. Macromolecules 2012, 45, 369. (18) Brás, A. R.; Pasquino, R.; Koukoulas, T.; Tsolou, G.; Holderer, O.; Radulescu, A.; Allgaier, J.; Mavrantzas, V. G.; Pyckhout-Hintzen, W.; Wischnewski, A.; Vlassopoulos, D.; Richter, D. Soft Matter 2011, 7, 11169. (19) Alim, K.; Frey, E. Phys. Rev. Lett. 2007, 99, 198102. (20) Tsolou, G.; Stratikis, N.; Baig, C.; Stehanou, P. S.; Mavrantzas, V. G. Macromolecules 2010, 43, 10692. (21) Milner, S. T.; Newhall, J. D. Phys. Rev. Lett. 2010, 105, 208302. (22) Rubinstein, M. Phys. Rev. Lett. 1985, 57, 3023. (23) Beaucage, G.; Kulkarni, A. S. Macromolecules 2010, 43, 532. (24) Sakaue, T. Phys. Rev. E 2012, 85, 021806.

4. CONCLUSIONS The static and dynamic properties of polymer systems are determined by many factors. One of the most important factors is the macromolecular architecture. Therefore, the twodimensional cyclic chains, especially “collapsed”, i.e., in the zero area within the chain contour, were the main subject of our interest. A coarse-grained lattice model of macromolecules was used and all atomic details suppressed: chains are represented as sequences of statistical segments. The system was at the highest possible density; i.e., all lattice sites were occupied by polymer segments or solvent molecules. The simplest possible force field consisted of the excluded volume only was introduced, and thus, the system was athermal modeling good solvent conditions. The sampling of the polymer conformational space was based on the conception of the dynamic lattice liquid (DLL). Therefore, we could study long-time dynamic properties of the system for polymer concentration between 0 and 1 which is impossible in the most of other simulation algorithms. Computer simulation studies were performed in order to determine static and dynamic properties of the model systems. The size and shape parameters show the change in the macromolecular structure with the chain internal architecture: the rings were considerably smaller and more spherical than their linear counterparts of the same length. Moreover, the scaling behavior of parameters describing “collapsed” cyclic chain size was found different from that of linear chains and close to that of lattice animals, i.e. the scaling exponent is located between 1 and 1.3. It was also shown that both lattice algorithms working at the highest possible density, i.e., the cooperative motion algorithm and the dynamic lattice liquid model, give the same results concerning static properties of polymer systems. The diffusion of macromolecules, which is crucial to their viscoelastic and rheological properties, was also I


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(25) Kim, J. U.; Yang, Y. B.; Lee, W. B. Macromolecules 2012, 45, 3263. (26) Cates, M.; Deutsch, J. J. Phys. (Paris) 1986, 47, 2121. (27) Bensafi, A.; Maschke, U.; Benmouna, M. Polym. Int. 2000, 49, 175. (28) Casassa, E. A. J. Polym. Sci., Part A 1965, 3, 605. (29) Wei, G. Chin. J. Polym. Sci. 1995, 13, 204. (30) Bishop, M.; Ceperley, D.; Frisch, H. L.; Kalos, M. H. J. Chem. Phys. 1980, 72, 3228. (31) Bishop, M.; Saltiel, C. J. J. Chem. Phys. 1986, 85, 6728. (32) Yethiraj, A. Macromolecules 2003, 36, 5854. (33) Adamczyk, P.; Polanowski, P.; Sikorski, A. J. Chem. Phys. 2009, 131, 234901. (34) Pakula, T. Makromol. Chem. Rapid. Commun. 1988, 9, 617. (35) Pakula, T. J. Chem. Phys. 1991, 94, 2104. (36) Shaffer, J. S. J. Chem. Phys. 1995, 103, 761. (37) Brown, S.; Szamel, G. J. Chem. Phys. 1998, 109, 6184. (38) Suzuki, J.; Tekano, A.; Deguchi, T.; Matsushita, Y. J. Chem. Phys. 2009, 131, 144902. (39) Sikorski, A. Polymer 1994, 35, 3792. (40) Wittmer, J. P.; Meyer, H.; Johner, A.; Obukhov, S.; Baschnagel, J. J. Chem. Phys. 2013, 139, 217101. (41) Fuereder, I.; Zifferer, G. J. Chem. Phys. 2011, 135, 184906. (42) Zifferer, G.; Preusser, W. Macromol. Theory Simul. 2001, 10, 397. (43) Brown, S.; Szamel, G. J. Chem. Phys. 1998, 108, 4705. (44) Müller, M.; Wittmer, J. P.; Cates, M. E. Phys. Rev. E 1996, 53, 5063. (45) Lang, M.; Fischer, J.; Sommer, J.-U. Macromolecules 2012, 45, 7642. (46) Vettorel, T.; Grosberg, A. Y.; Kremer, K. Phys. Biol. 2009, 6, 025013. (47) Deutsch, J. M. Phys. Rev. E 1999, 59, 2539. (48) Suzuki, J.; Takano, A.; Matsushita, Y. J. Chem. Phys. 2011, 135, 204903. (49) Suzuki, J.; Takano, A.; Deguchi, T.; Matsushita, Y. J. Chem. Phys. 2009, 131, 144902. (50) Hedge, G. A.; Chang, J.; Chen, Y.; Khare, R. J. Chem. Phys. 2011, 135, 184901. (51) Hur, K.; Jeong, C.; Winkler, R. G.; Lacevic, N.; Gee, R. H.; Yoon, D. Y. Macromolecules 2011, 44, 2311. (52) Brown, S.; Lenczycki, T.; Szamel, G. Phys. Rev. E 2001, 63, 052801. (53) Rubio, A. M.; Á lvarez, G.; Freire, J. J. Polymer 2008, 49, 628. (54) Bishop, M.; Michels, J. P. J. J. Chem. Phys. 1986, 84, 444. (55) Pakula, T.; Jeszka, K. Macromolecues 1999, 32, 6821. (56) Bishop, M.; Michels, J. P. J. J. Chem. Phys. 1986, 84, 447. (57) Bishop, M.; Michels, J. P. J. J. Chem. Phys. 1985, 82, 1059. (58) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. J. Chem. Phys. 2011, 134, 204904. (59) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. J. Chem. Phys. 2011, 134, 204905. (60) Kanaeda, N.; Deguchi, T. J. Phys. A: Math. Theor. 2008, 41, 145004. (61) Müller, M.; Wittmer, J. P.; Cates, M. E. Phys. Rev. E 2000, 61, 4078. (62) Suzuki, J.; Takano, A.; Matsushita, Y. J. Chem. Phys. 2008, 129, 034903. (63) Witz, G.; Rechendorff, K.; Adamcik, J.; Dietler, G. Phys. Rev. Lett. 2011, 106, 248301. (64) Witz, G.; Rechendorff, K.; Adamcik, J.; Dietler, G. Phys. Rev. Lett. 2008, 101, 148103. (65) Durbe, F.; Alim, K.; Witz, G.; Dietler, G.; Frey, E. Nano Lett. 2010, 10, 445. (66) Kosmas, M. K. Macromolecules 1990, 23, 2061. (67) Stratouras, M.; Kosmas, M. Macromolecules 1992, 25, 3307. (68) Zhang, L.; Xia, A.; Xu, Y. Eur. Polym. J. 2000, 36, 847. (69) Sikorski, A. Macromol. Theory Simul. 2001, 10, 38. (70) Sikorski, A. Macromol. Theory Simul. 2002, 11, 359.

(71) Camacho, C. J.; Fisher, M. E.; Singh, R. R. P. J. Chem. Phys. 1991, 9, 5693. (72) Oh, Y.; Cho, H. W.; Kim, J.; Park, C. H.; Sung, B. J. Bull. Korean Chem. Soc. 2012, 33, 975. (73) Camacho, C. J.; Fisher, M. E. Phys. Rev. Lett. 1990, 65, 9. (74) Reiter, J. Macromolecules 1900, 23, 3811. (75) Ż erko, S.; Polanowski, P.; Sikorski, A. Soft Matter 2012, 8, 973. (76) Mueller-Plathe, F. J. Chem. Phys. 1992, 96, 3200. (77) Pakula, T.; Teichmann, J. Mater. Res. Soc. Symp. Proc. 1997, 455, 211. (78) Pakula, T. J. Mol. Liq. 2000, 86, 109. (79) Polanowski, P.; Pakula, T. J. Chem. Phys. 2002, 117, 4022. (80) Pakula, T. Simulation Methods for Polymers. In Kotelyanskii, M., Theodorou, D. N., Eds. Marcel Dekker: New York and Basel, 2004. (81) Stauffer, A.; Aharony, A. Introduction to Percolation Theory; Taylor and Francis: London, 1994. (82) Hsu, H. P.; Nadler, W.; Grassberger, P. Comput. Phys. Commun. 2005, 169, 114. (83) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (84) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Claredon Press: Oxford, 1986. (85) Wen, P.; Zheng, N.; Li, L.; Li, H.; Sun, G.; Shi, Q. Phys. Rev. E 2012, 85, 031301. (86) Rudnick, J.; Gaspari, G. J. Phys. A: Math. Gen. 1986, 19, L191. (87) Nelson, H.; Hatton, T. A.; Rutledge, G. C. J. Chem. Phys. 1997, 107, 1269.

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Dynamic Properties of Linear and Cyclic Chains in Two Dimensions

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