1. Introduction

Biopolymers with a related architecture are also abundant in nature;for example proteoglycans[8] or the aggrecan molecules kept responsible for the very good lubricating properties in human joints[9]. In thiscontext the change in the solvent conditions is an important parameter, and theinfluence of these parameters on bottle-brush polymers has beenstudiedfor the case of single[10-12] and two-component[12-14] bottle brushes with a rigid backbone,suggesting in agreement with theoretical predictions structures rangingfrom individual collapsed chains at low grafting densities to the so-called “pearl-necklace” structures for intermediate densities andto homogenous cylinders and Janus-like structures at even higher densities.

Another interesting discussion regards the local "stiffness" traditionally measured by the persistence length l_p and the effective contour length[15-18].It is argued that finding a unique persistencelength measuring the “intrinsic” stiffness of a polymer cannotbe defined in the standard fashion with definitions that would all agree for Gaussian chains. Therefore, it has recently been shown that the persistence length depends not onlyon the backbone length[19,20], but on the solvent conditions as well [15]. Although there exist many experimental and theoretical studies for the linear dimensions of these macromolecules in various solvents[3,4,19,21-36],there are very few systematic studies of this problem[15], wherethe power laws and the associated effective exponents have been discussed.It has been shown that for bottle brushes with a flexible backbone evenat the theta point the side chains are considerably stretched,their linear dimension depending on the solvent quality only weakly, whilethe dependence of the persistence length on backbone length and temperature has alsobeen discussed[15].

The present work intends to make a contribution in giving a geometrical intuitionof single-component bottle-brush polymers with a flexible backbone undertheta and good solvent conditions and how the distribution of monomers changesby changing the various parameters. A pertinent discussion for a method to compute the persistence lengthof these complex macromolecules and its drawbacks will be also presented.The rest of this paper is laid out as follows. In Section 2,the current simulation model and its analysis are unfolded. A relevantdiscussion for the peculiarities of our system of interest is also provided. Section 3 presents a brief discussion of properties and our numerical results.This manuscript closes in Section 4, with a short summary.

2. Model and Methods

We describe the backbone chain and the side chains by a bead-spring model[37-43] where all beads interact with a truncated and shifted Lennard-Jones (LJ) potential U_LJ(r) and nearest neighbours bonded together along a chain also experience the finitely extensible nonlinear elastic potential U_FENE(r), r being the distance between the beads. Thus,

(1)

wherer_c= 2.5σ_LJ. The constant C is defined such that U_LJ(r = r_c) is continuous at the cut-off. Henceforth, units are chosen such that, σ_LJ = 1, the Boltzmann constant k_B = 1, and in addition also the mass m_LJ of beads is chosen to be unity. The potential of Eq. (1) acts between any pair of beads, irrespective of whether they are bonded or not. For bonded beads additionally the potential U_FENE(r) is used,

(2)

where the standard choice of parameters[40] (r₀ = 1.5 and k= 30) was adopted, and U_FENE (r> r₀)= ∞.Note that in our model there is no difference in interactions,irrespective of whether the considered beads are effective monomersof the backbone or of the side chains, implyingthat the polymer forming the backbone is either chemicallyidentical to the polymers that are tethered as side chains tothe backbone, or at least on coarse-grained length scales asconsidered here the backbone and side chain polymers are nolonger distinct. There is also no difference between the bondlinking the first monomer of a side chain to monomer ofthe backbone and bonds between any other pairs of bondedmonomers. Of course, our study does not address any effectsdue to a particular chemistry relating to the synthesis of thesebottle-brush polymers, but as usually done[40,44,45], we address universal features of the conformational properties of thesemacromolecules.

There is one important distinction relating to our previouswork[10-14] on bottle-brush polymers with rigid backbones: followingGrest and Murat[40], there the backbone was taken as aninfinitely straight line in continuous space, thus allowingarbitrary values of the distances between neighbouring graftingsites, and hence the grafting density σ could be continuouslyvaried. For the present model, where we disregard anypossible quenched disorder resulting from the grafting process, ofcourse, the grafting density σ is quantized: we denote here byσ=1 the case that every backbone monomer carries a sidechain, σ=0.5 means that every second backbone monomer hasa side chain, etc. Chain lengths of side chains were chosenasN=5,10,20, and 40, while backbone chain lengths were chosen as N_b= 50, 100, and 200, respectively.

It is obvious, of course, that for such short side chainlengths any interpretation of characteristic lengths in terms of power laws, such as, is a delicate matter,ν_eff being an effective exponent and characterizes only the specifiedrange of rather small values of N and not the limitconsidered by most theories[6,25,33-35]. Thus, the actual value of ν_eff is generally of limited interest, it only gives an indication to whichpart of an extended crossover region the data belong. However,we emphasize that: (i) our range of N nicely corresponds tothe range available in experiments[1,2,21-23,32,46-48] and (ii) the analysis in term of power laws with effective exponents isa standard practice of experimentalists in this context.A simulation analysis for the static properties of bottle-brushmacromolecules with flexible backbone have been discussed in detail previously[15].In this work we rather focus on the overall shapes that suchmacromolecules obtain, and discuss various aspects that couldnot be discussed in the frame of effective exponent analysis.

In our simulations, the positions of the effective monomers with label i evolve in time t according toNewton's equation of motion, amended by the Langevinthermostat[37-49]

(3)

whereU_i is the total potential acting on the i-th bead due to its interactions with the other beads at sites ,γ is the friction coefficient, and Γ_i(t) is the associated random force. The latter is related to γ by the fluctuation-dissipationrelation

(4)

Following previous work[37-49], we choose γ=0.5, the MDtime unit

(5)

beingalso unity, for our choice of units. Equation (3) was integrated using the leap from algorithm[50], with a time step, and utilizing the GROMACS package[51]. For the calculation of properties of the bottle brushes, typically 500 statistically independent configurations are averaged over. Of course, for bottle-brushes with large N_b equilibration of the polymer conformations is a difficult problem. Since we expect that end-to-end distance R_e and gyration radius R_g of the whole molecule belong to the slowest relaxing quantities, the judgment of the quality of results was based on the autocorrelations function of either of these quantities[15].

3. Results and Discussion

The interplay of various length scales in bottle-brush macromolecules results in interesting structures. The most extreme cases for the present system are shown in Fig. 1. For small values of N and temperatures close to theta (T = 3.0), the molecule can adopt conformations like the one of Fig. 1a, or that of Fig. 1b where still locally the backbone holds a high local flexibility. Moreover, at higher temperatures (e.g., T = 4.0, Figs. 1c and d) the whole macromolecule stretches due to the high affinity with the solvent molecules (Figs. 1c and d). Due to the longer side chains (N = 40, Fig. 1c) the backbone end beads are now not able to come close to each other due to the presence of the side chains which stretch the backbone monomers in the directions parallel to the backbone ends. It is clear that the side chains cause a significant stiffening of the backbone, at least on a coarse-grained scale, and that bottle brushes where N_b is not very much larger than N look like wormlike chains. It has been shown that, for temperatures close to the theta temperature, bottle-brush molecules can be very well described by the Kratky-Porodmodel[15], which describes the crossover from rods to Gaussian chains. Such analysis was based on the discussion of bond orientationcorrelations along the backbone beads, the measurement of the end-to-end distance and the use of effective exponents[15].

Here, we show plots (Figs. 2 -5) of an alternative definition of a “local” persistence length l_p(k) with the bond vector α_k connecting monomers at positions r_k and r_k-1 (α_k = r_k– r_k-1)[16,30,52]

(6)

but in the case of SAW chains[52]

(7)

However, only the pre-factor c can be taken as a measure of intrinsic stiffness, but not l_p(k) itself, since l_p(k) exhibits maximum at N_b / 2 which diverges to infinity proportional to as [4,52]. Furthermore, this problem is not improved when one considers an average of l_p(k) along the chain[5,16], while no divergence occurs for l_p(1)[16]. However, the use of l_p(1) is inconvenient in simulations due to the limited statistical accuracy. However, it is interesting to see that this definition in agreement with previous work[15] gives distinct dependence of the “persistence length” both on N_b and thesolvent quality.

Figure 1. (Colour online) Selected snapshot pictures of equilibrated configurations of bottle-brush polymers. Backbone monomers (when visible) are displayed in yellow (light grey) colour, side chain monomers in blue (darker grey). Cases a and b refer to σ = 0.5, Nb=100, N = 5, T = 3.0. Cases c and d refer to σ = 1.0, Nb= 100, N = 40, T = 4.0. Cases a, b, c, and d give a good idea of the range of structures one obtains for bottle-brush polymers with flexible backbone under theta and good solvent conditions

In the case of σ=0.5 (Figs. 2 and 3) the data are rather exhibit some noise due to the lower “local” stiffness of the backbone. As discussed, one could define a “persistence length” from the maximum of the curvesthat corresponds exactly to the centre of the chain (k / N_b = 0.5). It can be seen that for all N the increase of the temperature increases the stiffness ofthe backbone. Increase of σalso increases considerably thebackbonestiffness (Figs. 4 and 5). The persistence length depends on N, N_b, σ, and T.

The resulting estimates for a persistence length do not only depend on side chain length N and grafting density σ, but also on backbone chain length N_b and on temperature T making difficult a consistent analysisfor this quantity. However, when onestudies the variation of the end-to-end distance of the backbone[15] for T= T_Θ, an analysis in terms of the Kratky-Porodwormlike chain model becomes feasible. But in this case one must identify the contour length L implied by this model with the “chemical” contour lengthL_ch=N_bl_b, where l_b is the actual bond length, but ratherone has L=N_bl_b^eff with l_b^eff distinctly smaller than l_b. This effect results from the flexibility of the backbone on small scales; onlyon the scale of several backbone bonds does the stiffening due to the mutualside chains repulsions come into play. Thus, at the theta point both an effective contour length L and a persistence length l_p(k) are well-defined quantities, in terms of a fit of the data to the Kratky-Porod model, while under goodsolvent conditions such an analysis is not appropriate. In summary,an increase of σ, N, and T results in an increase of the effective persistence length along the backbone. Effects of the N_b willbe better described below.

Figure 2. (Colour online) Local persistence length lp(k) plotted versus k / Nb for σ = 0.5, Nb = 100, and T = 3.2 for different chain lengths N, as indicated

Figure 3. (Colour online) Local persistence length lp(k) plotted versus k / Nb for σ = 0.5, Nb = 100, and T = 4.0 for different chain lengths N, as indicated

Figure 4. (Colour online) Local persistence length lp(k) plotted versus k / Nb for σ = 1, Nb = 50, and T = 3.2 for different chain lengths N, as indicated

Figure 5. (Colour online) Local persistence length lp(k) plotted versus k / Nb for σ = 1, Nb = 50, and T = 4.0 for different chain lengths N, as indicated

Figure 6. (Colour online) Average dihedral angle along the backbone measured in Radians plotted versus N for σ = 0.5 (open symbols) and σ=1.0(full symbols) for differentNb and T, as indicated

An estimate of the local stiffness of the backbone could be providedby the average value of dihedral angles formed by four consecutive backbone beads and averaged for all possible such dihedral angles along the backbone, which is shown in Fig. 6.It is seen that increasing the grafting density σ, curves are considerably shifted to higher values,while the effect of temperature in the range of T = 3.0 (close to theta solvent) to T= 4.0 (good solvent) is rather smaller. Moreover, for N in the rangethere is a pronounced increase in D, which is mediatedfor N=40. Data also show a systematic increase in D as N_b increases showing that the local stiffness also clearly on average depends on N_b.

Figure 7. (Colour online) The mean square gyration radius of the side chains versus Nb for different σand T. Full symbols refer to data with N=40, while open symbols to data with N=20

As the persistence length clearly depends on N_b, one could gothe other way around and see if properties of the side chainsdepend correspondingly on the backbone length N_b. In Fig. 7we plot the mean square gyration radius of the side chains asa function of N_b. While dependencies on N, T, and σbecome apparent and they have been discussed in previous workin terms of effective exponents[15],it is rather difficult to extract a dependenceon N_b. Similar plots we have obtained for other properties ofthe sided chains showing the same behaviour with N_b. It would be interesting to simulate very longbottle brushes, but equilibration problems already impose a limitin the current study, while our result indicate that N_b = 200 is alreadyin the regime where such dependence enters a plateau-like regime.

Therefore, a more interesting discussion of the dependence on N_b may focuson the distribution of the monomers, which may indeed reveal some differences between the different cases.Thus,we turn our discussion to the overall shape of bottle-brushpolymers considering properties such as the asphericity andthe acylindricity[43].We follow the description of Theodorou andSuter[53] to define these quantities.Then, the gyration tensor reads as

(8)

whereS_i= col(x_i,y_i,z_i) is the position vector ofeach bead, which is considered with respect to the centre of massof the beads, and theover-bars denote an average over all beads N_ξ. When the gyration tensor of the whole chain is considered, thenN_ξ=nN+N_b, where nis the number of the side chain grafted onto the backbone monomers. For the side chains N_ξ = N, the gyration tensor and the properties are calculated for each side chain separately and thenan average over all results for each side chain is taken, whilefor the distribution of the backbone beads simply N_ξ = N_b. The gyration tensor is symmetric with real eigenvalues and a Cartesian system that this tensor is diagonal can always be found,

(9)

where the axes are also chosen in such way that the diagonal elements (eigenvalues of s) are in descending order

(10)

Theseeigenvalues are called the principal moments of the gyration tensor. From the values of the principal moments, one definesquantities such as the asphericity b,

(11)

When the particle distribution is spherically symmetric or has atetrahedral or higher symmetry, then b = 0. Theacylindricity c

(12)

is zero when the particle distribution approaches acylindrical symmetry. Therefore, the acylindricity and asphericityare relevant quantities that would describe some geometricalaspects of the monomer distribution in bottle-brush polymers.These quantities are taken with respect to s that is to the sum of the eigenvalues, i.e., the square gyration radius of the chain,which we also have calculated independently on our original Cartesian coordinate system in order to check our results. Subscript “s” to quantities b and c are referring to the side chains, “b” to the backbone beads, while b and c without subscripts refer to the distribution of allbeads belonging to the bottle-brush macromolecule.

Figure 8. (Colour online) Asphericity of the backbone beads versus N forσ=1.0 (full symbols) and σ=0.5 (open symbols) andvarious cases ofNb and T as indicated

Figure 9. (Colour online) A corresponding plot for the asphericity of the whole bottle-brush molecules is shown

Then, in Fig. 8 the asphericity for the distribution ofthe backbone monomers for various cases is shown. This distribution deviatesconsiderably from a spherical or higher symmetry as expectedfor bottle-brush macromolecules. For σ = 1.0 this effectis more pronounced compared to the case σ= 0.5. Also,increase of the side chain length N and of the temperature increases the values of b_b / s.On the other hand, when one takes into account all the monomersfor the calculation of asphericity (Fig. 9), a very small variation with N for σ = 1.0 is observed, while for σ=0.5 thebottlebrush obtains as a whole higher symmetrical structures.This is explained from the fact that the bottlebrush is an elongated object and the increase in the number of side chain monomers favors a spherical or higher symmetry for the whole bottle brush. The effect of temperature is similar.The increase of the backbone length N_b in the results of Figs. 8 and 9 leads to the increase of asphericity for the range of side chain lengths N we have consideredin our study.

Now if one measures b_s/ s for each side chain individually and averageover all side chains and plot this data versus N_b gets Fig. 10.Side chains have clearly symmetry close to a spherical one especiallycompared to the results of Figs. 8 and 9 and the interestingpoint is that the side chains adopt conformations with higher symmetry as the backbone length N_b increases. It might suggest that as N_bexceeds a certain value this dependence should disappear.We should mention here that for effects dueto the backbone ends are smeared out. Also, the results for N = 20 exhibit higher values of normalized asphericity than those for the case N = 40, where the chain is more stretched inthe radial directions from the backbone.

Figure 10. (Colour online) Asphericity of the side chains versus Nbis plotted for two different side chain lenghts N = 20 (full symbols) and N = 40 (open symbols) for different σand T as indicated

In Figs. 11-13 results for the acylindricity of the distribution of the backbone and the side chain monomers is shown. Overall, all distributions shown in Figs. 8-10 validate a cylindrical symmetry for all cases (backbone beads,all beads and the side chain beads). For the distribution ofbackbone monomers (Fig. 11) the increase of temperature T, the backbone length N_b, and the side chain length N obviously favours a more cylindricalsymmetry of the backbone monomers. The latter is also true for thedistribution of all bottle-brush monomers, but now a more favourable cylindrical symmetry for lower values of N_b is noted.

Figure 11. (Colour online) Acylindricity of the backbone beads versus N for two different temperaturesT=3.0 (full symbols) and T=4.0 (open symbols)for σ = 1.0 and differentNbas indicated

Figure 12. (Colour online) The acylindricity for all the bottle-brush beads is shown for σ = 1.0 and two different Nb, namely Nb = 100 (full symbols) and Nb = 50 (open symbols) for various temperatures as is shown on the plot

Figure 13. (Colour online) Acylindricity of the side chains versus Nb for the case N = 20 and different grafting density σand temperature T, as indicated

For N_b = 100 thebottlebrush is able to obtain configurations such as this of Fig. 1c, while for N_b = 50 the length of the backbone is rather small for the macromoleculeto allow for such curved structures since the persistence length along the bottlebrushis rather high. The distribution of side chain monomers tends to higher cylindrical symmetryas the grafting density σincreases. In this case also we see that a more cylindrical symmetry is favoured by increasing N_b for the range of values we have studied here. It would be interesting to simulate longer bottle brushes in order to find the limit of c_c / s as a function of N_b, but this is not possible with the simulation method we have adopted in this study. We note here that although a change of< R_g,s²> with N_b was not seen in Fig. 7, the distributionof the side chain monomers changes as it shown in Figs. 10 and 13 obtaininga more cylindrical shape with the increase of the backbone length. One should, however, also keepin mind that the number of side chains that are in the middle of the backbone is higher as thebackbone length increases, and these side chains are more stretched in the radial directions from the backbone, suggesting that the higher cylindrical symmetry with increased N_b can be also attributed to this reason.

4. Conclusions

In summary, we have demonstrated that the interplay of solvent quality, grafting density, side chain and backbone length in bottle brushes gives rise to very rich structural properties, where the distribution of monomers exhibit differences between bottle brushes, whereas other quantities would not reveal any dependence. A coarse-grained bead-spring model for bottle-brush polymers was studied via molecular dynamics, by varying both the chain length N_b of the backbone and the side chains N, for two values of the grafting density, under variable solvent conditions. The main target of the present work was to emphasize this geometric description of the shape of bottle-brush macromolecules at temperatures close to theta solvent conditions and in the good solvent regime. A short discussion on a way to extract the persistence length in bottle brushes and its relevant drawbacks was also briefly discussed for the present model.

ACKNOWLEDGEMENTS

P.E.T. would like to thank Profs. K. Binder and W. Paul for an exceptional collaboration over the last years. This work has benefitted by their insight and discussions. He also acknowledges financial support by the Austrian Science Foundation within the SFB ViCoM (Grant No. F41).N.G.F. has been partly supported by MICINN, Spain, through Research Contract No. FIS2009-12648-C03.