The influence of flexible side chains on the dimensions of flexible polymers

CTPS 100

Computational and Theoretical Polymer Science 9 (1999) 307–326

The influence of flexible side chains on the dimensions of flexible polymers J.T. Wescott, S. Hanna* University of Bristol, H.H.Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK Received 13 October 1998; received in revised form 9 March 1999; accepted 10 March 1999

Abstract Molecular dynamics simulations of single chains in vacuo have been used to investigate the role of flexible branches on the dimensions and in particular the backbone rigidity of flexible polymers. A series of isotactic poly-a-olefin chains was chosen as a model system to study because of their simplicity of structure. Q-conditions were simulated for each branched chain by careful adjustment of the Van der Waals parameters, and properties such as end-to-end length, radius of gyration and persistence length were measured. As expected the backbone rigidity of the polymers was found to be determined by a balance between the coiling of the backbone induced by a gauche state at the branch point and the size of the side chain. Trends in persistence length with branch content compared favourably to the previous Rotational Isomeric State and experimental measurements of the characteristic ratio. The stiffening of the backbone under good solvent conditions was also investigated and it was found that the persistence length could be systematically increased by increasing the size of the side group, with the shape of the side group also making a small difference to the sampled backbone conformations. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Molecular dynamics; Persistence length; Poly-a-olefin chains; Q-condition

1. Introduction The versatile properties of polymers, be it mechanical, morphological or thermal, are a direct consequence of the preferred conformations of individual chains and hence understanding the influence of chemical structure on chain conformations is of great interest. For example, it is well known that the physical properties of flexible linear polymers can be modified through the introduction of side groups. The expected effect of the introduction of side groups is to cause the polymer backbone to kink. In stereoregular chains this can result in the formation of helices, such as those found in the crystalline forms of poly-aolefins with short side chains, but in an entangled melt this will usually lead to enhanced flexibility. Conversely, there are also instances where it has been suggested that the addition of large side groups will result in a loss of flexibility. For example, the poly(di-n-alkylsiloxane)s form liquid crystalline mesophases whose stability increases with the length of the alkyl side chains [1]. Also, poly(phenyl vinylene) derivatives show an increase in the conjugation length as the length or bulk of the side group * Corresponding author. Tel.: 1 117-928-8771; fax: 1 117-925-5624. E-mail address: [email protected] (S. Hanna)

increases [2]. In both cases, the measured properties are found to be enhanced by increased backbone rigidity. One further example is that of the poly-a-olefins [3], in which the side chains have a profound effect on the glass transition temperature Tg. Longer linear side chains systematically decrease Tg whilst the introduction of additional methyl substituents on the linear side groups, forming bulky side chains, dramatically increases Tg (see Table 1). This variation in Tg may be directly related to the mobility and flexibility of the chain, which is a function of the side chain content. A similar correspondence between the side branches and the flexibility of various poly(methacrylate)s has been discussed in the review of Xu et al. [4]. They noted that the characteristic ratio, C∞, is systematically increased for chains with larger bulkier side groups due presumably to the steric hindrance of the side chain causing an increase in kR 2l0. Saariaho et al. [5] recently used Monte Carlo (MC) simulations to study the influence of side chain topology in athermal dilute solutions for chains composed of beads of various diameters. An enhancement of the persistence length, a measure of the chain rigidity, was found for relatively short side chains. It is valuable to study the role of such excluded volume chains because they correspond experimentally to chains in good solvent conditions, but it is

1089-3156/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S1089-315 6(99)00021-5

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Table 1 Glass transition temperatures for some isotactic poly-a-olefins (as given by the Polymer Handbook) and list of molecules considered in this study Name

Abbreviation

Glass transition temperature, Tg (8C)

Polypropylene Polypentene-1 Polybutene-1 Polyhexene-1 Poly(4-methylpentene-1) Poly(3-methylbutene-1) Poly(3-methylpentene-1) Poly(3-methylhexene-1) Poly(4-methylhexene-1) Poly(5-methylhexene-1) Poly(3,3 0 -dimethylbutene-1) Poly(4,4 0 -dimethylpentene-1)

PP PT1 PB1 PH1 P4MP1 P3MB1 P3MP1 P3MH1 P4MH1 P5MH1 P33DMB1 P44DMP1

218 225 240 250 40 50

often more desirable to compare the properties of chains in an unperturbed state, i.e. at Q-conditions. This ensures that any change in the properties are due to a combination of the bonded arrangement, the hindrance to torsional rotation and any localised steric effects, and not just the excluded volume effects. The definition of a Q-chain suggests that our findings may apply to polymer chains in the melt or glassy state, as well as in a Q-solvent. The criterion for defining a Q-condition has been a topic of controversy for many years. One definition, based on experiment, is that in which the second virial coefficient of the osmotic pressure expansion, A2, tends to zero, i.e. there are no non-ideal contributions to the virial and the chain is unperturbed. A Q-solvent is defined as one in which A2 ˆ 0 and corresponds to a situation in which neither the polymer–solvent nor the polymer–polymer contacts are favoured over the other, and the chain is neither expanded nor collapsed. Another method, in broad agreement with this, is that in which the scaling of chain dimensions with molecular weight by the universal exponent n is used to define the solvent quality. A Q-state is defined as that in which n becomes equal to 1/2, a situation that also arises for an ideal random walk. Above the Q-temperature, the solution is in a good solvent regime and n becomes about 0.6. These arguments are based solidly on scaling theory, and very accurate simulation studies have revealed exponents of 0.5 and 0.588 [6] for random and self-avoiding walks, respectively. The scaling of n ˆ 1/2 also implies that the chains adhere to Gaussian statistics. However, the exact equivalence of these definitions has been rejected by theory [7] and recent simulations [8] suggest that residual three body effects are evident when n ˆ 1/2, requiring logarithmic corrections of the Q-state. It is clear that the situation is more complicated when attempting to define a unique Q-condition for regularly branched chains (often referred to as polymer combs in literature). Lipson [9] has considered corrections to the

scaling laws for excluded volume chains, but not for chains in the Q-state. In the case of very long chains, we expect that the branches (at least for the short branch comb-like polymers under consideration here) would not effect the scaling laws, i.e. n should still be 1/2 at the Q-point. Indeed the result of Forsman [10] predicts that a branched chain can be considered to be exactly linear, provided it is sufficiently long. However, the effect that branches may have on the virial coefficients and hence the equivalence of the Q-state definitions is not clear. The approximation of an unperturbed state by Q-conditons is brought further into doubt by observations of the so-called “specific solvent effects” [11] in which the local chain conformations are influenced by the size and shape of the solvent. It appears a little strange to assign a special status to a Q-state when the dimensions of a given Q-chain are dependent on the particular solvent used. There are a large number of simulation studies relating the solvent quality of a system to the chain dimensions, mostly pertaining to linear polymers [12–16]. To obtain the Q-conditions in the absence of explicit solvent or a surrounding polymer matrix it is necessary to adjust the energy terms to compensate for the extra intrachain interactions that occur. One way of achieving this is by simply ignoring the interactions between atoms or monomers separated by more than five bonds along the chain. This has been successfully applied to linear polyethylene [16] and biphenol-A-polycarbonate [17], where it has been shown that the simulated chain dimensions are in good agreement with experimental data. This method has the advantage of being computationally inexpensive although some differences are found in the shapes of the coil and in the endto-end distance distributions when compared to Q-chains generated by other means. An alternative method is to scale the attractive interactions between the widely separated segments so as to cancel exactly the excluded volume effect. This has the advantage of retaining all interactions which may otherwise be ignored by an ad hoc truncation, and which might be important for the local conformations. The scaling of the strength of the interaction can be justified as a mean-field screening effect of either the solvent or the surrounding polymer. This has been successfully achieved in a number of lattice MC simulations where the attractive interactions are simply scaled between 0 and 1 and the excluded volume is generated by the single occupancy of a lattice site. Similarly, Sariban et al. have also used continuum MC simulations to define a Q-point for linear polyethylene [16], in which the attraction is determined by a direct scaling of the Van der Waals potential. The aim of the present study is to use computer simulation to observe the effect of a systematic variation of the side chain structure on the backbone conformations, and hence on the resulting dimensions and flexibility of a range of branched (comb-like) polymer chains. A scheme similar to that described above was used to allow us to explore

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326 Table 2 The potential functions and DreidingII force-field parameters used Type

Function

Parameters

Bonds Angles

Constrained Ka …a 2 a0 †2

˚ (C–C) ˆ 1.54 A Ka ˆ 50 kcal mol 21 rad 22; a 0 ˆ 109.4718 Ku ˆ 1.0 kcal mol 21 see Table 3

Dihedrals Lennard-Jones

Ku …1 1 cos 3u† D0 ‰…R0 =rij †12 2 2…R0 =rij †6 Š

both Q and good solvent conditions using molecular dynamics (MD) simulations of atomistic systems. The careful determination of the Q-point for each type of comb was an inevitable consequence of these aims, and contributed to our understanding of the mechanisms controlling the chain flexibility and average dimensions in such systems. 2. Methods 2.1. Materials Polyethylene (PE) and isotactic poly-a-olefin chains were chosen for simulation as these molecules have a simple architecture that can be compared easily to a range of theoretical, simulation and experimental systems. The models consisted of short chains of united CH, CH2 and CH3 units with terminal CH3 groups. The monomer for the poly-aolefin chains is [–CH2CHR –]n where the R group is an nalkyl or branched alkyl group. PE, PP, PB1 and PT1 (see Table 1) chains of backbone length N ˆ 10, 25, 50 (and N ˆ 100 where possible) were simulated in the full range of solvent regimes. To gauge the effect of excluded volume only, a range of highly branched N ˆ 50 length isotactic poly-a-olefins were also simulated under good solvent conditions. These are also listed in Table 1. 2.2. Solvent conditions To obtain an unperturbed chain in atomistic simulations the size of the Van der Waals interaction was scaled in order to balance the effects of the excluded volume and attractive interactions. The non-bonded interactions were given by VLJ ˆ

A B 2g 6; 12 rij rij

…1†

where A ˆ 41 s 12, B ˆ 41 s 6, 1 and s are the standard Lennard-Jones parameters and g is the scaling factor for

309

the attractive part of the potential. The excluded volume of the chain is generated mostly by the r 212 factor and hence remains almost constant. One way of characterizing the effect of this scaling is by defining an effective well depth c through Eq. (2)

cˆ

g2 1 ; NA kB T

…2†

where g 21 is the scaled LJ well depth VLJ, T is the temperature, NA and kB are the Avogadro and Boltzmann constants, respectively. As we were unable to calculate the virial expansions from single chain simulations, the definition of solvent conditions must come entirely from the scaling of the mean square endto-end distance, kR 2l, or the mean square radius of gyration, kS 2l, with chain length (/ N 2n ). This is a well-discussed topic in theory and simulation, with the Q-condition being the state in which the scaling exponent recovers that of a random walk (n ˆ 1/2). The size of the coil is of course not the same as that from a random walk of the same length, as real chains cannot intersect and have bond angle constraints. However, the comparison of the two unperturbed states, as embodied in the characteristic ratio, gives one useful measure of the flexibility. The characteristic ratio CN: CN ˆ

kR2N21 l0

…3†

…N 2 1†`2

will be independent of N at Q-conditions and in the limit of N ! ∞. Determining the Q-conditions is possible by plotting CN versus N as for the unperturbed chain this quantity will show a well-defined plateau as CN tends asymptotically to C∞. The Q-point may also be determined by plotting kR 2l/ (N 2 1) or kS 2l/(N 2 1) versus c for chains of various lengths as this quantity will lose its N-dependence at the point where the curves cross. Both this method and that using CN were used in this study. 2.3. Force-field The DreidingII force-field parameter set [18] was implemented in its standard form except for (1) the 1–4 interactions, which were excluded for these simulations so that a direct comparison with the MC studies of Sariban et al. could be made [16], and (2) the use of constrained bonds so that a larger timestep could be employed. DreidingII was chosen primarily for its united atom parameterisation and simple potential functions. These include harmonic

Table 3 Non-bonded/Lennard-Jones parameters for the DreidingII force-field (These parameters were also used in the repulsive-only non-bonded potential of Eq. (4)) United atom

˚) R0 (A

D0 (kcal mol 21)

˚ 12) A (10 6 kcal mol 21 A

˚ 6) B (kcal mol 21 A

CH CH2 CH3

3.9828 4.0678 4.1526

0.0946 0.1984 0.2500

2.3386 4.0713 6.5659

1171.4 1797.5 2563.1

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bond-angle bending terms, a simple cosine dihedral potential and a Lennard-Jones (LJ) non-bonded function. The functions and parameters are given in Tables 2 and 3. The form of the dihedral angle potential is particularly important for the overall dimensions of the chain and a range of dihedral potential forms are available. For simulations of sp 3 carbon chains, the preferential sampling of trans and gauche states over all others is the important feature, which is generated by the cos3u dependence. For simulations of PE, however, it was shown that in order generate the proportions of trans and gauche found experimentally a potential with modified barrier heights is necessary. This potential was not used for simulations of PE because of the desire for a parameter set that was fully transferable to all branched models. The same cos 3u potential was used for all the dihedral angles defined by the backbone and the side branches. A corollary to this is that we will not expect to predict exact dimensions for the chains under consideration. We will, however, hope to see the correct trend in these values. The existence of branches along the backbone means there are CH groups at the branch points and CH, CH2 and CH3 groups in the side chains. The LJ12-6 interactions for each of these groups is different but the same scaling g 2 was applied to the well depth 1 of all united atom types. The non-bonded interaction parameters used are given in Table 3. Simulations were also run comparing models in which only the excluded volume interactions were considered. Rather than using a large scaling of the well depth, these comparisons were made using an athermal (repulsive only) non-bonded interaction. The form of this potential is given in Eq. (4) and the parameters used are given in Table 3    rij ; …4† VLJ ˆ D0 exp h 1 2 R0 where h ˆ 12. All the non-bonded potentials were implemented with a ˚ . This relatively small cut-off radius has sharp cut-off of 6 A been employed in other polymer simulations [19] and was chosen to reduce the amount of CPU time dedicated to the non-bonded interactions. 2.4. Simulation conditions The trajectories of the MD runs were computed using a standard velocity Verlet algorithm with a 2.0 × 10 215 s time step. The relatively large time step was possible due to the exclusion of the fastest molecular motions through the use of united atoms and constrained bonds (constrained using the SHAKE algorithm). The dl_poly [20] program was used to run the simulations on an Origin 200 silicon graphics computer. Constant-NVE MD simulations were carried out, although the temperature was maintained at 400 K during the equilibration via periodical scaling of the individual

atom velocities. This temperature was chosen because it is close to the experimental Q-temperatures of polyethylene [21] and polypropylene [22] in biphenyl (127.5, 1258C). The temperature was tracked throughout the run to ensure that there was no systematic drift from this value. The atomic co-ordinates were saved at 1 ps intervals after equilibration. Several different starting configurations were used for each simulation to reduce the risk of dominant vibrational modes from biasing the statistics or of the simulation getting stuck in one region of phase space. This was especially necessary in the simulations of collapsed chains where the latter is a particular problem. Each sub-simulation was run for 2 ns and included a small equilibration period of 2 ps. One of the first checks for these simulations was whether the average property values had converged within the proposed simulation time, which was governed primarily by the total number of atoms in the simulation. Uniform sampling of the configurational phase space for a 50 unit PE chain was achieved in about 10 ns (10 000 configurations) but took more than 100 ns for the more complicated molecules and longer lengths. The long run time was necessary in order to sample as many configurations as possible. The statistics generated were comparable to the MC simulations on the same types of models [12–16]. With branched molecules, the total number of atoms increases rapidly with backbone length so that simulations quickly become cumbersome. For example, an N ˆ 50 backbone P4MP1 chain has 150 united atoms, requiring about 100 000 configurations or 100 ns simulation to achieve convergence of the property values. The length of simulation time required for the average properties to converge for systems changed 2 approximately as Ntot which places a limit on the usefulness of MD for isolated chain simulations. Brownian dynamics is one method by which the phase space might be sampled more effectively. In this method the chain is subject to random momentum changes which simulate collisions of the chain with solvent molecules, allowing quicker relaxation of, and conversion between, the chain conformations. However, to obtain the substantial increase in efficiency required to sample the conformations of chains longer than 100 backbone atoms, the use of MC methods is most appropriate. Nevertheless, as we will demonstrate, the chain dimensions of even relatively short chains (N , 50) are affected by chain topology and the trends found can be assumed to be applicable to much longer molecules. 2.5. Analysis of trajectories The calculated ensemble average properties were the mean square end-to-end distance kR 2l, the mean square radius of gyration, kS 2l, the characteristic ratio, CN, the persistence length, `pers, the distribution of backbone dihedral angles and the shape of the coil as measured by the asphericity, Asph, and acylindricity, Acyl, of the coil.

311

Persistence length

N

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lpers

20

40

60

80

100

number of backbone atoms N

Fig. 1. Plot of kqlN versus N for an ideal flexible chain. The persistence length `pers is given by the plateau value.

The persistence length, `pers, is a quanitiy used by the liquid crystalline polymer community to describe the stiffness or rectilinearity of a chain. Several different models can be used to define `pers including the worm-like chain model of Kratky and Porod [23]. However, the most direct definition, the projection of the end-to-end distance on to the direction of the first bond (see Eq. (5)), is easily implemented in molecular simulations:

3.1. Defining the Q -point using the collapse transition In general, as with other studies determining the Qpoints, the collapse transition was characterised by plotting (a) 18

…5†

14

The value of interest occurs when the quantity kqlN is no longer dependent on N, i.e. `pers ˆ limN!∞ kqlN . The value `pers is then determined from a plot of kqlN versus N by the projection of the plateau back to the ordinate axis (see Fig. 1). Following Tanaka and Mattice [24], the asphericity and acylindricity were computed using the principal components of the mean square radius of gyration S2x ; S2y and S2z , where S2x . S2y . S2z (see Eqs. (6) and (7))   1 2 kSy l 1 kS2z l =kS2 l; Asph ˆ kS2x l 2 2

(b)

12 10 8 6 4 2 0 0.01

0.1

ψ

1

N = 100 N = 50 N = 25 N = 10

2.5

…7†

It is expected that in an expanded state the chain is more cylindrical and less spherical. For a rigid rod, acylindricity is zero and asphericity is near 1. For spheres, both acylindricity and asphericity are near 0. The backbone dihedral angles were plotted as a frequency histogram with a bin width of 18 and were compiled using all available backbone torsions within the model for each configuration. The trans state was defined as that for which the dihedral angle is 1808.

10

3

…6†

/ (N - 1)

  Acyl ˆ kS2y l 2 kS2z l =kS2 l:

N = 100 N = 50 N = 25 N = 10

16

/ (N - 1)

^ kqlN ˆ kRN · `l:

3. Polyethylene in various solvent conditions

2 1.5 1 0.5 0 0.01

0.1

ψ

1

10

Fig. 2. Plots of (a) kR 2l/(N 2 1) and (b) kS 2l/(N 2 1) versus c for PE chains of N ˆ 10–100.

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ψ = 0.0156 ψ = 0.0899 ψ = 0.1295 ψ = 0.1405 ψ = 0.1599 ψ = 0.2023 ψ = 0.2254 ψ = 8.9900

10 9

Characteristic Ratio CN

8 7 6 5 4 3 2 1 0 0

10

20

30

40 50 60 Backbone Length N

70

80

90

100

Fig. 3. CN versus N for PE (N ˆ 100) using various c values. CN values averaged over all sub-chains of the full chain.

kR 2l/(N 2 1) versus c (or kS 2l/(N 2 1) versus c ) as in Fig. 2. The N ˆ 25, 50 and 100 curves do not quite cross at a welldefined point indicating that a small N dependence remains, due probably to the short chain lengths used. As N increases, the crossover from the expanded coil to the collapsed state becomes sharper in both plots, as expected. The smearing out of the transition is most explicit for chains of length N ˆ 10 where the effect of changing the non-bonded interaction is minimal because the chain can rarely fold back upon itself. The N ˆ 50 and N ˆ 100 curves cross at c ˆ 0.17–0.18 and the N ˆ 50 and N ˆ 25 curves at c ˆ 0.20– 0.24, a range of c that is not in numerical agreement with Sariban et al. [16] (c , 0.07), even after adjustment for the different LJ parameters used. The differences between the studies lie in the molecular detail of this atomistic simulation, by the use of more than one atom type, and more ˚ was used here as opposed critically, the cut-off radius (6 A ˚ used by Sariban et al.). The smaller cut-off dramato 12 A tically reduces the number of pair-wise interactions of each chain unit (i.e. each has a lower effective co-ordination number) and hence extra “attraction” per chain unit is required to compensate and recover the Q-conditions. This amounts to using a higher c value in the simulation. This reasoning is also consistent with the larger value of c 0 , 0.214 obtained in lattice MC simulations [25] where Table 4 Comparison of 1 0 values for different length poly-a-olefins (errors ^ 0.02) Length

PE

PP

PB1

PT1

100 50 25

0.13 0.15 0.21

0.20 0.24 0.49

– 0.24 0.37

– 0.27 0.20

the effective co-ordination number of each unit is restricted to that of the lattice. For chains with backbone lengths less than 100 units, we were unable to define a unique c 0 for each chain type, and as we were unable to simulate longer chains, it was felt necessary to define a c 0 value for each chain length used. 3.2. Defining the Q -point using the characteristic ratio Adherence to the scaling laws for linear chains was achieved in general in the good solvent regime (n . 0.5) and in the Q-state (n ˆ 0.5). However, this was not observed for collapsed chains where there was a distinct non-linearity at high N. The non-linearity manifests itself as a dip in the plateau value of the CN versus N curve (see Fig. 3) which is just apparent for c ˆ 0.14. This dip was also observed by Cifra [26] in lattice MC simulations. The Qstate was defined for the value of c for which the dip just disappeared. For excluded volume chains the plateau values are significantly higher than those in the Q-state and as suggested by Fig. 2 the value of c 0 determined in this manner is dependent on the length of the simulated chain. The results are summarised in Table 4. A larger c 0 is required for small N because the likelihood of chain units, more than three bonds apart interacting is reduced by the persistence of direction of short chain lengths. Thus, to recover the Q-conditions, the interactions that do occur must be attributed extra “attraction” in the same way that a reduction in co-ordination number leads to an increase in c 0. 3.3. Persistence length measurements The effect of solvent conditions on the persistence length

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

ψ = 0.0156 ψ = 0.0899 ψ = 0.1296 ψ = 0.1405 ψ = 0.1599 ψ = 0.2023 ψ = 0.2254 ψ = 8.9900

10 9 Persistence Length / Angstroms

313

8 7 6 5 4 3 2 1 0 0

10

20

30

40 50 60 Backbone Length N

70

80

90

100

Fig. 4. kqlN versus N for PE (N ˆ 100) using various c values. kqlN values averaged over all sub-chains of the full chain.

measurements can be seen in Figs. 4 and 5. In Fig. 4 the variation of `pers with c is similar to that found for the characteristic ratio, although the plateau is not found at exactly the same c value. Where a distinct plateau was not apparent, the initial turnover of the curves was used to define `pers. Fig. 5 shows `pers as a function of c for N ˆ 50 ˚ in the excluded and N ˆ 100, where `pers ranged from 6.8 A ˚ in the collapsed regime. In the Qvolume regime to 2 A ˚ was assigned for the longest state, a value of 5.0 ^ 0.1 A (N ˆ 100) chain simulated. (A small N dependence remains ˚ greater than that for with this value, approximately 0.1 A

N ˆ 50). These values of `pers are smaller than those quoted experimentally and computationally determined values, ˚ . This which generally imply that `pers is greater than 6.6 A discrepancy is not surprising, given the nature of the forcefield, and it should be emphasised that these simulations were not designed to reproduce the experimental property values but are aimed to identify the trends in a model system. For a better correspondence with experiment, a dihedral potential such as that developed for n-alkanes [27] could be used along with a larger cut-off radius. This would undoubtedly lead to lower flexibility as a preference

7

N = 100

Persistence Length / Angstroms

N = 50 6

5

4

3

2 0.01

0.1

ψ

1

Fig. 5. Plot of persistence length versus c for PE; N ˆ 50 and N ˆ 100.

10

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Table 5 Ratios of principal components of kS 2l and asphericity and acylindricity values for PE, PP, PB1 and PT1 chains in the athermal, Q and collapsed regimes Type and length

State

c0

kSx 2l/kSz 2l kSy 2l/kSz 2l

Asph

Acyl

PE 100

Athermal Q Collapsed Athermal Q Collapsed Athermal Q Collapsed Athermal Q Collapsed

0 0.13 1.5 0 0.20 4.0 0 0.24 4.0 0 0.24 4.0

15.80 13.30 2.00 15.37 11.15 2.34 15.62 11.33 2.08 15.77 9.86 2.35

0.83 0.81 0.31 0.84 0.79 0.38 0.85 0.80 0.33 0.85 0.78 0.38

0.14 0.15 0.13 0.13 0.15 0.14 0.13 0.15 0.14 0.13 0.15 0.13

PP 100

PP 50

PB1 50

3.30 3.06 1.35 3.09 2.77 1.43 3.42 2.75 1.37 3.06 2.51 1.37

for trans states is built into the dihedral potential form. Both changes, however, would require a re-determination of the Q-state as c 0 is always model dependent. The large difference between the `pers values for the excluded volume chains and the collapsed chains emphasises the importance of quantifying the solvent conditions when comparing the persistence length values of different molecules. 3.4. Chain shape The asphericity and acylindricity values for PE in the three solvent regimes are given in Table 5. The difference in shape between the Q-chain and the athermal chain is small. The asphericity values for the athermal and Q-chains are much larger than the collapsed state as expected. In each case the asphericity of the athermal chain is higher than that 12

kR2 l=kS2 l: A further quantity of interest is the ratio kR 2l/kS 2l, as this also varies with solvent power and is N-dependent for the short chains under consideration here (see Fig. 6). For long Gaussian chains in the Q-state this quantity is theoretically equal to 6(N 1 1)/(N 1 2) but reduces to two for fully collapsed globules. These simulations show that the expanded coils have values of kR 2l/kS 2l consistently greater than six whilst the smallest value of kR 2l/kS 2l achieved was 3.4 with c ˆ 9.0, further evidence that the fully collapsed state was not sampled. The transition from the expanded to the collapsed coil is evident and the chain with kR 2l/kS 2l closest to six is consistent with assigning c 0 ˆ 0.13 for N ˆ 100. 4. Poly-a-olefins with small linear side chains It is clear from the simulations of PE that for the comparison of persistence lengths between architectures, the solvent regime must be carefully monitored. A similar Q-point analysis was carried out for three isotactic ψ = 0.0156 ψ = 0.0899 ψ = 0.1295 ψ = 0.2023 ψ = 0.2254 ψ = 8.9900 freely jointed chain model

10

Ratio of to

of the corresponding Q-chain. The acylindricity values show virtually no change across the solvent regime. These results are in good agreement with lattice simulations of Tanaka et al. [24] although these results show (1) slightly higher acylindricity values even for athermal chains, perhaps because the chains have more freedom off-lattice, and (2) higher asphericity values, a consequence of the relatively short chains under consideration here. The collapsed chains in these simulations do not extend as far into the collapsed regime as the MC simulations again because of the shorter chains used here

8

6

4

2

0

0

10

20

30

40 50 60 Backbone length N

70

80

90

Fig. 6. Plot of kR 2l/kS 2l versus N for PE (N ˆ 100) using various c values.

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(a)

18

N = 100 N = 50 N = 25 N = 10

16 14 / (N - 1)

315

12 10 8 6 4 2 0 0.01

(b)

0.1

ψ

1

3 N = 100 N = 50 N = 25 N = 10

2.5

/ (N - 1)

10

2 1.5 1 0.5 0 0.01

0.1

ψ

1

10

Fig. 7. Plot of (a) kR 2l/(N 2 1) and (b) kS 2l/(N 2 1) versus c for PP chains of various lengths.

poly-olefins with short linear side groups, namely polypropylene(PP), poly(butene-1)(PB1) and poly(pentene1)(PT1). These have one, two and three side chain atoms, respectively. 4.1. Determination of c 0 The collapse transition for PP, PB1 and PT1 is evident in Figs. 7–9. Again, a c 0 that is truly independent of N was not apparent in any plot. As a first estimate though c 0(PP) , 0.25, c 0(PB1) , 0.29 and c 0(PT1) , 0.36 might be assigned. Comparing this to c 0(PE) , 0.13, it appears that the presence of branches adds to the excluded volume of the chain and more attraction is necessary to

cancel out the effect. The difference between c 0(PE) and c 0(PP) is much greater than that between c 0(PP) and c 0(PB1) or c 0(PT1). Thus, it appears that the addition of the first side chain atoms has the greatest effect. This suggests that the branch point rather than the branch length has the greatest impact on the backbone conformations, at least for short linear-branch combs under Q-conditions. The determination of the Q-state using the plateau of CN versus N to determine c 0 for each chain length yielded the values given in Table 4. Fig. 10 shows that for different length PP chains, the same c value yields curves of different shapes for CN versus N, which confirms that, as with PE, c 0 is larger for shorter chains.

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J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

(a)

18

N = 50 N = 25 N = 10

16

/ (N - 1)

14 12 10 8 6 4 2 0 0.01 (b)

0.1

ψ

1

3 N = 50 N = 25 N = 10

2.5

/ (N - 1)

10

2 1.5 1 0.5 0 0.01

0.1

ψ

1

10

Fig. 8. Plot of (a) kR 2l/(N 2 1) and (b) kS 2l/(N 2 1) versus c for PB1 chains of various lengths.

4.2. Dependence of persistence length on solvent conditions and chain length Fig. 11 shows a plot comparing `pers values of PE, PP, PB1 and PT1 versus c for N ˆ 50 chains. In the collapsed regime, the branched chains appear to be slightly more flexible whilst in the excluded volume regime the branches increase `pers (and kR 2l or kS 2l). Fig. 12, a plot of `pers versus c for N ˆ 50 and N ˆ 100 PP chains, also shows a small N dependence for the persistence length which is slightly greater in the excluded volume regime than was apparent for PE (see Fig. 5). Table 6 gives a comparison between the persistence length values of Q-chains of various lengths. In the Q-state

PP is less rigid than PE of the same backbone length and PB1 is slightly more flexible again. However, the trend is reversed for PT1 with `pers greater than for PB1. Experimental determinations of persistence lengths of poly-aolefins are scarce in the literature and are limited to those inferred from C∞, mostly for atactic polymers, and inferred from intrinsic viscosity measurements. The inference of kS 2l and hence, C∞ from such experiments has been criticised by Zirkel et al. [11] as disagreement with SANS measurements have been found, particularly for the temperature coefficient of the unperturbed dimensions. Whilst intrinsic viscosity measurements predict a negative value for k ˆ d(lnkR 2l)/ dT for atactic PP and PB1, neutron scattering from samples in the melt state and in solution yield positive k .

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

(a)

18

Nevertheless, the values of C∞ for isotactic poly-a-olefins generated using a five-state Rotational Isomeric State (RIS) model were compared favourably to the experimental C∞ by Wittwer and Suter [28]. These results are used in Table 6 to imply the persistence lengths using Eq. (8). This is an approximation that only strictly holds for ideal Gaussian chains and therefore has limited applicability. It is used here only as a guide to the trends in persistence length as no better measure is available

N = 50 N = 25 N = 10

16

/ (N - 1)

14 12 10 8 6 4

C∞ ˆ

2 0 0.01

(b)

ψ

0.1

1

/ (N - 1)

10

3 N = 50 N = 25 N = 10

2.5 2 1.5 1 0.5 0 0.01

ψ

0.1

1

10

Fig. 9. Plot of (a) kR 2l/(N 2 1) and (b) kS 2l/(N 2 1) versus c for PT1 chains of various lengths.

2`pers 2 1; `

…8†

˚. where ` ˆ C–C, bond length ˆ 1.54 A Table 6 shows for all cases that PP is more flexible than PE. The MD results showed that PB1 is also more flexible than PP although the experimental and RIS measurements given in the Table show a slight increase in the persistence length. However, other measurements [29] yielded smaller values of C∞ for PB1. These imply estimates for `pers of 3.9–4.5 (RIS) and 4.9 (exptl.) in better agreement with the present results. However, it is clear that in all cases adding an ethyl rather than a methyl as a side chain has a smaller impact on the backbone conformations, and also that when the side chain gets long enough, as with PT1, there is an increase in `pers. The latter finding suggests that the use of longer or more bulky side branches may further increase `pers. The trends are apparent even for the N ˆ 25 length chains. It is also apparent that the solvent conditions play a large role in determining the balance between the extra excluded volume of the side group and the flexibility induced by the branch point.

8 ψ = 0.0156

7

ψ = 0.0624

Characteristic Ratio CN

6

ψ = 0.1481 5 ψ = 0.2023

4 3 2 1

ψ = 0.6393 0 0

10

20

317

30

40 50 60 Backbone Length N

70

80

90

100

Fig. 10. Comparison of CN versus N for various c interactions for PP chains of length N ˆ 100 and N ˆ 50.

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J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

PE (N=50) PP (N=50) PB1 (N=50) PT1 (N=50)

8

Persistence Length / Angstroms

Persistence Length / Angstroms

7

6

5

4

4.4 4.2 4 3.8 3.6 3.4 3.2 3 0.2

0.25

0.3 ψ

0.35

0.4

3

2 0.01

0.1

ψ

1

10

Fig. 11. Plot comparing persistence length values versus c for PE, PP, PB1, PT1 chains (N ˆ 50).

4.3. Torsional distributions Fig. 13 shows the distribution of backbone torsions for PE, PP and PB1 in the collapsed, Q and excluded volume regimes. The similarity between the Q-state and the excluded volume regime is quite apparent for PP and PB1 with gauche states changing to trans in the expanded chains. For PE there is slightly greater variation in the shape of the distributions about each peak. All the molecules in the Qstate and good solvent regime have three well-defined peaks corresponding to the trans (highest peak) and the two gauche states. This is despite the use of a dihedral potential

in which the trans and gauche are given equal weight. The symmetry of Fig. 13 (i.e. equal probability of G 1 and G 2 states) suggests that the backbone conformations in these vacuum simulations should be unaffected by the tacticity of the polymer. This was indeed observed, in simulations of syndiotactic and atactic PP where similar values for kR 2l, kS 2l and `pers were obtained to the isotactic case. These observations are undoubtedly a consequence of our use of a simplified force-field and will not necessarily be borne out experimentally. For the collapsed state, the behaviour is quite different. In PE the collapsed state shows loss of the gauche states and

7

N = 100

Persistence Length / Angstroms

N = 50 6

5

4

3

2 0.01

0.1

ψ

1

Fig. 12. Plot of persistence length values versus c for PP (N ˆ 50 and N ˆ 100).

10

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326 Table 6 Comparison of persistence length values from the current simulations with those from experimental and RIS calculations. The experimental and RIS data were calculated from C∞ values taken from Wittwer and Suter [28] ˚) (errors for MD simulations ^ 0.1 A Polymer

PE

PP

PB1

PT1

Experimental Theoretical (RIS) MD simulation

6.7 6.7 5.0 4.4 3.6

5.4–5.8 4.8 4.5 3.8 3.0

6.2 5.5 – 3.5 2.9

6.9 5.9 – 4.0 3.1

N ˆ 100 N ˆ 50 N ˆ 25

the appearance of states at dihedral angles around zero. The non-bonded energy dominates the dihedral angle energy term and possible conformations could include tight bends with short stretches of all trans states. Such arrangements are not possible in PP or PB1 because of the disruption caused by the branches. Instead a much wider range of dihedral angles is sampled. Again the dihedral energy term is dominated by the very ‘sticky’ non-bonded term but in a much less regular way than for PE. There is some evidence of preferred torsions for PB1 but further investigation is needed if the fluctuations in these distributions are to be related to the structure of the collapsed globule. 4.4. Chain shape The asphericity, acylindricity and principal components of kS 2l for PE, PP and PB1 are given in Table 5 in the three solvent regimes. Firstly, the asphericity and acylindricity values in each regime are approximately independent of the branch type and chain length (at least down to N ˆ 50). For athermal chains Asph ,0.84 and Acyl ,0.13, for Q-chains Asph ,0.80 and Acyl ,0.15 and for collapsed chains Asph ,0.35 and Acyl ,0.14. Secondly, the values of kS2x l=kS2z l and kS2y l=kS2z l for athermal and collapsed chains is again independent of the chain type and length. However, for Q-chains kS2y l=kS2z l reduces by 0.25 per branch atom added and kS2x l=kS2z l from 13.3 for PE to 11.2 for PP and 9.9 for PB1.

5. Branched poly-a-olefins in good solvent The role of the excluded volume of side branches in persistence length enhancement has been demonstrated in the above simulations and also by Saariaho et al. [5]. To further investigate this effect, simulations using a wide range of branch types were carried out. The good solvent conditions (athermal solvent) were ensured by using the repulsive-only non-bonded potential for all chain units. The molecules simulated included P3MB1, P4MP1, P5MH, P33DMB1 and P44DMP1. A summary of the structure of these molecules is given in Table 7. They were all of backbone length N ˆ 50 and each simulation was run for 100 ns.

319

5.1. Dependence of average dimensions on side branch content In the previous section it was shown that adding branches to excluded volume chains has the effect of increasing the average size of the coil. The scaling of kS 2l with N gave scaling not of n ˆ 0.588 but slightly greater than 0.6. Lipson [9], using a lattice MC technique on self avoiding long chain combs, found a need to include correction terms to the dominant scaling factor, n (see Eq. (9)) in order to get a better fit h i …9† kS2 l ˆ AN 2n 1 1 …B=N D † 1 …C=N† ; where A, B and C are amplitudes and n ˆ 0.588 and D ˆ 0.47 are universal scaling exponents. The data in the current simulations were not sufficiently accurate to warrant such an analysis and it is possible that the failure to achieve n ˆ 0.588 was due entirely to the use of short chain lengths (N ˆ 50). 5.2. Trends in persistence lengths The calculated persistence lengths of all the athermal branched molecules are shown in Fig. 14. The increasing length of side chain gives rise to a substantial increase in the persistence length. This is true for a series with both linear and branched side groups. Adding extra methyl groups to the linear side chains increases the persistence length further, although usually to a lesser extent than adding the extra atom, to form a longer side chain. One exception occurs for the addition of the extra methyl on P3MB1 to form P33DMB1. The extra crowding of the backbone ˚ increase caused by the second methyl gives rise to a 1 A in the persistence length. These observations might lead us to suggest that a polymer consisting of flexible units such as these simple hydrocarbons could form relatively stiff chains by (1) crowding the backbone with side chain units and (2) adjusting the solvent conditions. A molecule such as the poly(2-propylpentene-1) molecule shown in Fig. 15 under good solvent conditions might be a candidate for enhanced persistence ˚ was measured for a length and indeed a value of 14 A N ˆ 50 length molecule. This is not an accurate determination of `pers as in an N ˆ 50 length chain there will only be about four persistence lengths in the backbone, but it demonstrates the potentially large persistence length that could be attained with appropriate solvent conditions and side chain content. 5.3. Backbone conformations Fig. 16 shows the dihedral distributions for the three series. Firstly, considering the linear side chains, the effect of an ethyl side branch as opposed to a methyl is to decrease the proportion of trans states relative to the gauche. On further increasing the side chain length the proportion of

320

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326 0.012

PE

Fraction occuring per degree

0.01

(a)

Collapsed chain (ψ =0.9989) Theta chain (ψ =0.1295) Excluded volume (ψ =0.0000)

0.008

0.006

0.004

0.002

0

0

60

120

180 240 Dihedral angle / degrees

300

360

0.012

PP

Fraction occuring per degree

0.01

(b)

Collapsed chain (ψ =0.9989) Theta chain (ψ =0.2398) Excluded volume (ψ =0.0000)

0.008

0.006

0.004

0.002

0

0

60

120

0.012

Fraction occuring per degree

(c)

300

360

Collapsed chain (ψ =0.9989) Theta chain (ψ =0.2398) Excluded volume (ψ =0.0000)

PB1

0.01

180 240 Dihedral angle / degrees

0.008

0.006

0.004

0.002

0

0

60

120

180 240 Dihedral angle / degrees

300

360

Fig. 13. Plots of backbone dihedral angle distributions for (a) PE (N ˆ 100), (b) PP (N ˆ 100) and (c) PB1 (N ˆ 50) in collapsed, Q and excluded volume states.

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

domination of trans states. It would appear that the side chains interact so strongly with each other and the backbone that the backbone is forced away from its preferred trans state to approximately ^ 108 from this. This must be due to the extreme crowding of the P33DMB1 side chains around the backbone. The asymmetry of the curve may suggest that the conformational space has not been sampled fully, but nevertheless the occurrence of peaks away from the trans position is something unique to this molecule.

Table 7 Summary of a series of poly-a-olefins Side chain length

1 2 3 4

Number of CH2’s in side chain (m)

0 1 2 3

Terminal group CH3

(CH3)2

(CH3)3

PP PB1 PT1 PH1

– P3MB1 P4MP1 P5MH1

– P33DMB1 P44DMP1 –

321

gauche states stay approximately constant whilst the trans state actually increases again slightly. These results show that the initial effect of adding branches is to increase the proportions of gauche states but that the excluded volume of larger side branches begins to take effect when the side chain is a propyl group or longer. The results also show that the dihedral distribution need not directly imply the persistence length as the trend in `pers would perhaps predict increasing numbers of trans states as the side chain length was increased. The second series (Fig. 16(b)) reveals how the addition of branches to the side chains again influences the backbone. The effect of adding one extra methyl to the side chain is to reduce the proportion of trans and increase the proportion of gauche states. Another extra methyl added further increases the proportion of gauche states but has a smaller effect on the trans peak. Again this is contrary to the generalisation that more gauche states implies a more coiled chain. In the third series, similar comments apply to the addition of one methyl to the PB1 side chain. However, upon addition of the second methyl a complication prevents the

5.4. Varying the side chain shape A number of simulations were run to see if the arrangement of atoms in the side group was important in determining the persistence length. Simulations for two series of poly-a-olefin chains (pentenes and hexenes) with a single extra methyl attached to the linear side chain were run. (see Fig. 17). It should be noted that the number of side chain atoms within each series is constant. Fig. 18 shows that the effect of positioning the methyl at different points along the side chain is negligible for the hexene series whilst for the pentene series a small but significant increase in the stiffness can be found by moving the methyl closer towards the backbone. The trends in these simulations show qualitative agreement with the variations in the experimentally determined values of C∞ for poly(methacrylate)s [4] insofar as flexibility and the centre of mass of the side chain are factors in determining backbone flexibility. Figs. 19 and 20 show plots of the persistence length and the expansion coefficients versus the total number of side chain atoms for all the excluded volume chains. It is clear that the number of side chain atoms is the dominant factor in determining the

10 9.5

Persistence Length / Angstroms

9 P5MH1

8.5

P44DMP1 P33DMB1

8

PH1 P4MP1

7.5 P3MB1

7 6.5

PT1

PB1 PP

6 5.5 5 0

1

2 3 side chain length

4

5

Fig. 14. Plot of persistence length versus length of side chain for three series with different terminal groups. All chains had repulsive-only non-bonded interactions.

322

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

˚. Fig. 15. Section of a poly(2-propylpentene-1) molecule. The persistence length under good solvent conditions was about 14 A

stiffness of the backbone, whilst the arrangement of the atoms within the side chain is of lesser importance.

6. Discussion In this work the role of side groups in perturbing the trajectory of the backbone in a polymer chain has shown that (1) in the good solvent regime, the persistence length is enhanced as the side chain grows in length and bulk, and (2) in the Q-solvents it is the branch point that is most important with the influence of the excluded volume of the side groups cancelled to a large extent by the extra attraction needed to recover Q-scaling. This was despite the observation that the chain shape and backbone conformations are very similar

for excluded volume chains and Q-chains. The lack of numerical agreement with experimental persistence lengths (as calculated from C∞) is a direct and understandable consequence of the force-field used. However, the trends of the model systems are clear. In the good solvent regime, the branches play a large part in enhancing the chain rigidity which helps to explain the increase in mesophase stability shown by the poly(di-nalkylsiloxane)s [1]. These polymers, however, were in the melt-state that leads us to infer that the chains in a meltmesophase are far from the Q-state. This assumption is not unreasonable given the aligned nature of the chains in a mesophase. In other words, it is plausible to suggest that the chains in a mesophase melt do not obey Gaussian statistics, due to the co-operative nature of the liquid crystal

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

323

PP (N=50) PB1 (N=50) PT1 (N=50) PH1 (N=50)

0.014

(a)

Fraction occuring per degree

0.012

0.01

0.008

0.006

0.004

0.002

0

0

60

120

180 240 Dihedral angle / degrees

300

360

PT1 (N=50) P4MP1 (N=50) P44DMP1 (N=50)

0.014

(b)

Fraction occuring per degree

0.012

0.01

0.008

0.006

0.004

0.002

0

0

60

120

180 240 Dihedral angle / degrees

300

360

PB1 (N=50) P3MB1 (N=50) P33DMB1 (N=50)

0.014

(c)

Fraction occuring per degree

0.012

0.01

0.008

0.006

0.004

0.002

0

0

60

120

180 240 Dihedral angle / degrees

300

360

Fig. 16. Plot of distribution of dihedral angles with variation of (a) linear side chain length (PP, PB1, PT1, PH1), (b) side chain bulk (PT1, P4MP1 and P44DMP1), (c) side chain bulk (PB1, P3MB1 and P33DMB1) for N ˆ 50 length chains. All chains had repulsive-only non-bonded interactions.

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J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

3

5

2

4

1

(a)

(c)

(b)

3

5 6

2

4

1

(d)

(f)

(e)

Fig. 17. Illustration of the position of extra methyl group along the side chain in P3MP1, P4MP1, PH1, P3MH1, P4MH1 and P5MH1.

behaviour. Also, the possibility of the formation of mesophases in inherently flexible polymers is very real if chains with suitably long (and/or bulky) side chains can be dissolved in a good solvent. The definition of the Q-state demonstrated that a greater Lennard-Jones attraction was required to compensate for the extra excluded volume of the side chain as more atoms were added to the side branch. The resulting trends in flexibility, namely that the PP, PB1 and PT1 were all less stiff than PE, was in agreement with some RIS calculations of the unperturbed dimensions [28-29]. This is despite the fact that the RIS calculations do not directly involve any non-bonded interactions. The influence of the side groups was inferred instead through the weighting factors used in the RIS procedure that were based on a priori energy minimisations of

short lengths of chain. The increased rigidity of PT1 over PB1 that was observed here for Q-chains suggests that bulkier branches, e.g. those in P4MP1, will result in persistence lengths larger than for PE. The investigation of such effects will require the use of a more efficient sampling technique such as can be achieved with MC, and will be the subject of future work. An additional point is that the generation of persistence lengths and characteristic ratios involved calculations using sub-chains of the polymer, in order to obtain the maximum statistics possible from one simulation. It has been suggested that this approach may influence the end result because the inner sub-chains are slightly expanded relative to the outer sub-chains due to their more restricted range of motions. This idea has been tested for excluded volume

9

Persistence Length / Angstroms

8.75

P4MH1 P5MH1 P3MH1

8.5 8.25

P3MP1 PH1

8 P4MP1

7.75 7.5 7.25 7 2

3 4 5 Position of extra methyl group in monomer

6

Fig. 18. Plot of persistence length of backbone versus position of the branch methyl in the side chain for N ˆ 50 chains. All chains had repulsive-only nonbonded interactions.

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

325

9 P4MH1 P5MH1 P3MH1 P44DMP1

Persistence Length / Angstroms

8.5

P3MP1 P33DMB1 / PH1

8

P4MP1 7.5 PT1 P3MB1 7

PB1

6.5 PP

6

5.5 0

1

2

3 4 Number of side chain atoms

5

6

Fig. 19. Plot of persistence length of the backbone versus the total number of side chain units for N ˆ 50 chains. All chains had repulsive-only non-bonded interactions.

chains [30] and was found to be in agreement with the renormalisation group theory predictions of this effect. For chains near Q-conditions, however, it has been shown that the perturbations from Gaussian behaviour [31] are small and hence should have little effect on the observed trends. One parameter that might be expected to effect the persistence length values, although not the trends in `pers is the choice of non-bonded cut-off radius. A change in the cut-off radius shifts the position of the collapse transition as

expected, but the overall effect on the persistence length trends is hard to quantify and needs to be investigated further. Further improvements in the development of forcefields for branched alkanes would also provide us with results that are more readily comparable to experimental findings. Of all the parameters, it is undoubtedly the choice of dihedral potential, which has the most influence on the chain stiffness. Therefore, refinement of this

Expansion factor / PE

1.6

P5MH1

1.5 P33DMB1 PH1

1.4

P44DMP1

P4MP1 1.3

PT1 P3MB1

1.2 PB1 1.1

PP PE

1 0

1

2 3 4 Number of side chain atoms

5

6

Fig. 20. Plot of the linear expansion factors kR 2l/kR 2lPE for the N ˆ 50 poly-a-olefin chains. All chains had repulsive-only non-bonded interactions. The ratios kS 2l/kS 2lPE were also calculated and found to be coincident with the data plotted here.

326

J.T. Wescott, S. Hanna / Computational and Theoretical Polymer Science 9 (1999) 307–326

aspect of the force-field will be necessary if we wish to reproduce the experimental persistence lengths and tacticity effects. For example, one possibility would be the use of the united atom force-field specially developed for simulations of liquid branched alkanes [32]. However, the principal aim in the present study was to investigate persistence length changes in model branched systems, where the selection of such a relatively complex force-field was not deemed appropriate. 7. Conclusions Single chains of PE, PP, PB1 and PT1 were simulated under a range of solvent conditions and it was found that for a given Lennard-Jones well depth the presence of branches affects the solvent conditions experienced by the backbone. Thus to recover Q-scaling for the main chain it was necessary to increase the well depth in order to cancel the effect of the extra excluded volume introduced by the short linear branches. In the resulting Q-state the persistence lengths of PP, PB1 and PT1 were all found to be smaller than for the same length of PE chain, the addition of the first branch atom having much more influence than the second and subsequent atoms. Consideration of the asphericity and acylindricity showed that the shape of the coil in the athermal, Q-state and collapsed state was independent of the side branch type, although a reduction in the ratios of the principal components of kS 2l was observed. The enhancement of persistence length upon addition of the branch atoms is evident in the good solvent regime where the excluded volume of the side chains serves to expand the main chain and increases the persistence length. It is most notable that the trend of increased flexibility of PP and PB1 in the Q-state is reversed in the good solvent regime. However, simulations of the less flexible PT1 chains suggest that `pers is enhanced in both regimes when bulkier side groups are present. A range of excluded volume poly-a-olefins were simulated which showed that both the branch length and the bulk increased the rigidity of the main chain but also that, paradoxically, the presence of bulkier side groups slightly increased the number of gauche states in the backbone. Acknowledgements The authors would like to thank the EPSRC for supporting this project through a PhD studentship. They are also

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