The competition of hydrogen-like and isotropic interactions on polymer collapse
aa r X i v : . [ c ond - m a t . s o f t ] J un The competition of hydrogen-like and isotropic interactions onpolymer collapse
J. Krawczyk , A. L. Owczarek and T. Prellberg ∗ Department of Mathematics and Statistics,The University of Melbourne,Parkville, Victoria 3052, Australia. School of Mathematical SciencesQueen Mary, University of LondonMile End Road, London E1 4NS, UKNovember 4, 2018
Abstract
We investigate a lattice model of polymers where the nearest-neighbour monomer-monomerinteraction strengths differ according to whether the local configurations have so-called“hydrogen-like” formations or not. If the interaction strengths are all the same then theclassical θ -point collapse transition occurs on lowering the temperature, and the polymerenters the isotropic liquid-drop phase known as the collapsed globule. On the other hand,strongly favouring the hydrogen-like interactions give rise to an anisotropic folded (solid-like) phase on lowering the temperature. We use Monte Carlo simulations up to a length of256 to map out the phase diagram in the plane of parameters and determine the order ofthe associated phase transitions. We discuss the connections to semi-flexible polymers andother polymer models. Importantly, we demonstrate that for a range of energy parameterstwo phase transitions occur on lowering the temperature, the second being a transition fromthe globule state to the crystal state. We argue from our data that this globule-to-crystaltransition is continuous in two dimensions in accord with field-theory arguments concerningHamiltonian walks, but is first order in three dimensions. ∗ email: [email protected],[email protected],[email protected] Introduction
The self-avoiding walk (SAW) on a lattice [1, 2] is a key model in statistical mechanics for thestudy of the static properties of polymers. Incorporating interactions in this model make itpossible to represent many features of real polymers. Regardless of the constraints of the latticerelative to the real world, the model mimics very well many properties of physical systems [3].The self-avoiding walk on a lattice is a random walk which is not allowed to visit a lattice sitemore then once. Each visited lattice site is considered as a monomer of the polymer chain. Acommon way [2] to model intra-polymer interactions in such a walk is to assign an energy to eachnon-consecutive pair of monomers lying on the neighbouring lattice sites. This is the canonicalISAW model which is the standard model of polymer collapse using self-avoiding walks. Withthis modification one studies a polymer in a solvent, where the energy between monomers canbe attractive or repulsive and depends on temperature. If the energy is repulsive the polymerbehaves as a swollen chain (the so-called excluded-volume state) regardless of temperature andone says that it is in a good solvent. When the energy is attractive, and the temperature islow enough, the chain becomes a rather more compact globule [2, 4], reminiscent of a liquiddroplet: this is also known as the poor solvent situation. The transition point between thosetwo phases is called the θ -point; it is a well studied continuous phase transition (see [5] andreferences therein).The modelling changes as soon as we want to describe any biological system (e.g. proteins), inwhich the hydrogen bonding plays an important role [6]. One of the main features of the bondingis that the interacting residua lie on a partially straight segments of the chain. Hydrogen-likebonding was first modelled on the cubic and square lattices using Hamiltonian paths by Bascle etal. [7]. A monomer acquires a hydrogen-like bond with its (non-consecutive) nearest neighbourif both of them lie on straight sections of the chain (see Figure 2). The interacting self-avoidingwalk modified to have only such interactions will be referred to as the hydrogen-like bondingmodel, or rather hb -model. The hb -model was studied in mean-field approximation [7] and afirst-order transition from a high-temperature excluded-volume (swollen) phase to a quasi-frozensolid-like phase was found in both two and three dimension. Hence this would indicate that itis a different type of transition from the θ -point. Note also that the low temperature hb -phaseis anisotropic whereas the collapsed globule of the standard θ -point model is isotropic. The hb -model on the square lattice was studied directly by Foster and Seno by means of the transfermatrix method [8] and by Krawczyk et al. [9] on both the square and cubic lattice using a MonteCarlo method. In both studies a first-order transition was found between an excluded-volume(swollen-coil) state and an anisotropic ordered compact phase in two and in three dimensions,again in opposition to the θ -point [2].It is appropriate to compare this difference between the behaviour of the hb -model and2 -point models with the difference between the behaviour of interacting semi-stiff polymersand the fully-flexible θ -point polymers. This is because hydrogen bonding induces an effectivestiffness in the polymer between those monomers that are taking part in the interactions. As thetemperature is lowered the proportion of the monomers experiencing this stiffness increases sowhile not all the segments of the polymer feel this stiffness at high temperatures, the proportionof monomers involved with nearest-neighbour hb -interactions increases towards unity as thetemperature is lowered. In three dimensions, Bastolla and Grassberger [10] discussed so-calledsemi-stiff self-avoiding walks, which interact via all nearest-neighbours, as in the θ -point model,and include a bending energy. They showed that when there is a strong energetic preference forstraight segments, this model undergoes a single first-order transition from the excluded-volumehigh-temperature state to a state similar to the low-temperature solid-like state of the hb -model.Intriguingly, if there is only a weak preference for straight segments, the polymer undergoes twophase transitions: on lowering the temperature the polymers undergoes the θ -point transition tothe liquid globule followed at a lower temperature by a first-order transition to the frozen phase.We should point out though that in two dimensions the transition between the globule and thefrozen state has only been studied in Hamiltonian walks, and there it seems to be continuousone [11].To complicate matters further, there is at least one other model using a different definitionof interactions which could be regarded as hydrogen-like bonding. This model [12] definesinteractions between parallel segments, that is, bonds of the lattice occupied by the walk andso connecting monomers, see Figure 2. We will call this model the interacting bond model.Studying this model by means of Bethe approximation, Buzano and Pretti [12] found, in bothtwo and three dimensions, two phase transitions: while decreasing the temperature the θ -collapseto isotropic globule phase is followed by a first order transition to a solid-like phase. Hence,this is similar to the semi-stiff model for weak stiffness. The interacting-bond model in twodimensions has recently been studied by Foster [14] and also displays two transitions. In [13]Buzano and Pretti added isotropic nearest-neighbour monomer-monomer interactions to theinteracting-bond model, and investigated the phase diagram in three dimensions, again in theBethe approximation. They showed that the phase diagram is similar to the interacting-bondmodel. However, if the interaction between monomers are repulsive, there is only one first-orderphase transition from the swollen coil to the solid-like phase. This is again reminiscent of thesemi-stiff model for strong stiffness.It is therefore of some interest to study an enhanced hb -model where non-hydrogen bond near-est neighbours are also considered. Hence, in this paper we investigate a model of self-interactingself-avoiding walks with two types of nearest-neighbour interaction: the hb -interactions andnearest-neighbour interactions that are not hydrogen bonds, which we denote as nh -interactions.The competition between these two types of interaction ( hb versus nh ) leads to a three-phase3hase diagram, with excluded volume, globule and frozen phases. Of special interest is the com-parison to the semi-stiff model. One key question is whether there can exist two phase transitionson lowering the temperature. The order of the transitions in two and three dimensions are alsoof interest. We use a Monte Carlo technique, known as FlatPERM [15], to study self-avoidingwalks on the simple cubic and square lattices with interactions as described.The paper is organised as follows. In Section 2 we explain more carefully details of themodel. In Section 3 the phase diagram in both dimensions is discussed. A discussion of theanisotropy of the model is also given. We conclude with a summary and discuss the similarityof this model to the semi-stiff model. The polymer is modelled on a square and simple cubic lattice as a self-avoiding walk withinteractions between different types of nearest-neighbours monomers: that is monomers thatare not consecutive in the walk but nearest neighbours on the lattice. The strength of theinteraction depends on the relative position of the monomers involved in the interaction tothose next to them on the walk. A segment is defined as a site along with the two adjoiningbonds visited by the walk, and we say that a segment is straight if these two bonds are aligned.The ‘hydrogen bonds’ are nearest neighbour interactions that are between monomers where bothare part of straight segments of the polymer, see Figure 1.Figure 1: The types of nearest-neighbour interactions between two straight segments of thepolymer involved in the hb -interactions: parallel segments (left) and orthogonal segments (right).In the model studied in this paper these two types are weighted equally. In two dimensions,only parallel interactions are possible.Our model weights both the parallel and orthogonal hb interactions equally. Our model alsoincludes all other possible nearest neighbour interactions and assigns them a different Boltzmannweight. The two types of distinguished interactions are shown in Figure 2. As just describedinteractions between monomers sitting on straight lines, as in monomer − ε hb . The other kind of interaction consist of any nearest-neighbour interaction between non-consecutive monomers that are not hydrogen bonds, as in the pairs 1-6 and 10-13 for example.The energy of those interactions is denoted as − ε nh .
13 42 567 8 9 10 111213141516
Figure 2: Interactions in 3 d interacting self-avoiding walk. In our model we distinguish two differ-ent kind of interactions, which are denoted by two different colours. The red colour (dark shad-ing) denotes hydrogen-like interactions ( hb ) (monomers 2-5, 5-14) whereas the green colour (lightshading) denotes all other interactions between two neighbouring monomers ( nh ) (monomers1-6 and 10-13 for example).Note that the number of all nearest-neighbours interactions, m , is equal to the sum of thenumber of the two types of interaction considered in our model, that is m = m nh + m hb . Theenergy of configuration ϕ n of an n -step walk is calculated as E ( ϕ n ) = − m hb ( ϕ n ) · ε hb − m nh ( ϕ n ) · ε nh , (2.1)where m hb and m nh are the number of hydrogen-like bond interactions and non hydrogen-likenearest-neighbour interactions, respectively. The inverse temperature is denoted as β = 1 /k B T ,where k B is the Boltzmann constant and T the absolute temperature. We define for convenience β hb = βε hb and β nh = βε nh . The partition function is then given by Z n ( β hb , β nh ) = X m hb ,m nh C n,m hb ,m nh e β hb m hb + β nh m nh (2.2)with C n,m hb ,m nh the density of states. Canonical averages are calculated with respect to thisdensity of states.Since we will consider simulational results along lines in parameter space at a constant ratioof ε nh /ε hb = β nh /β hb we define ε nh = γ , and ε hb = 1 − γ , so that the energy is then given by E = − m hb · (1 − γ ) − m nh · γ. (2.3)5n our study we will analyse the (reduced) specific heat to investigate the phase diagram: thatis, C ( T ) = 1 T h E i − h E i n . (2.4)We will also consider fixing one of the parameters either β hb or β nh and varying the other.To analyse the possible phase transitions we then use the fluctuations in number of monomersof appropriate type. In the case of β hb being constant we consider σ ( m nh ) = h m nh i − h m nh i . (2.5)When β nh is constant we consider σ ( m hb ) = h m hb i − h m hb i . (2.6)Simulations have been performed with FlatPERM algorithm [15]. We have simulated themodels using a two parameter implementation (utilising m hb and m nh ) for length n = 128 wherethe simulation directly estimates this density of states C n,m hb ,m nh . We have also performed oneparameter ( m hb or m nh ) simulations for systems of size 256 where the simulation estimatespartial summations of this density of states over one of the variables. When γ = 1 / θ -transition from coil to globule state. The θ -transition is a second-order phasetransition in both two and three dimensions. In two dimensions the established crossover expo-nent is φ = 3 / α = 2 − /φ = − / θ -transition, the specific heat is expected to diverge loga-rithmically [18]. The collapsed state is an isotropic dense liquid-like droplet with a well-definedsurface tension [19, 20].On the other hand, for γ = 0 the model becomes the hb -model studied by Foster and Seno[8] on the square lattice and Krawczyk et al. on the square and simple cubic lattices [9]. In bothtwo and three dimensions, there is a single first-order transition to a folded crystalline state,which is anisotropic.Using these results as a starting point, one therefore expects there to be at least these threephases (swollen, globule, crystal) in the full two-dimensional parameter space explored here.For γ > /
2, the hb -interactions are suppressed relative to the non- hb interactions, and thesimplest hypothesis would be that the θ -transition is not affected. To test this hypothesis, we6ave considered the line γ = 1 below. For γ close to zero, the hb -interactions dominate, and onemay expect that there exists some range of values for γ for which the first-order transition ofthe pure hb -model persists. To test this, we need to consider a small value of γ . We of coursethen need to consider other values of γ to see if the two transitions can occur for fixed γ andwhether there exist any other phases. To gain an understanding of which values of γ we may need to consider more closely, we firstexamine the fluctuations in the numbers of interaction across a wide range of ( β hb , β nh ). Asin previous work [21, 22, 9], we found the use of the largest eigenvalue of the matrix of secondderivatives of the free energy with respect to the parameters β nh and β hb most advantageousto show the fluctuations in a unified manner. Figure 3 displays density plots of the size offluctuations for − . ≤ β nh , β hb ≤ . β nh and β hb , we expect themodel to be in the excluded volume universality class of swollen polymers, since at β nh = β hb = 0the model reduces to simple self-avoiding walks. For fixed β hb and large β nh , we deduce that thepolymer is in a globular state, while for fixed β nh and large β hb , we deduce that the polymer isin the anisotropic crystalline state.A fixed value of γ corresponds to a straight line of slope (1 − γ ) /γ on the plots in Figure 3.The line with γ = 0 . γ ≥ . θ -transition. Below we consider theline γ = 1 to verify this. The line with γ = 0 . γ around 0 (slope sufficiently large) for which the system willundergo a single first-order hb -model like transition on lowering the temperature. The figuresalso suggest that there exists a critical value of γ , say, γ c , where this scenario ends.Therefore one may deduce that there exists some range of γ between γ c and 0 .
5, not neces-sarily the whole range, for which the system undergoes two transitions on lowering the temper-ature. In this region, the polymer starts in the swollen state at high temperatures, undergoesa θ -transition to a globular state on lowering the temperature, and on lowering the tempera-ture further, undergoes a further (novel) transition to the crystalline state. We verify this byconsidering the line with γ = 0 . n = 128 at γ = 0 . , .
4, and1 . γ = 0 . γ = 1 .
0, there is one peak in the specific heat7igure 3: Density plots of the logarithm of the largest eigenvalue of the matrix of second deriva-tives of the free energy with respect to β nh and β hb (the lighter the shade the larger the value).Both plots are for n = 128, for two- and three-dimensional systems, top and bottom, respectively.The lines shown indicate cross-sections for which we have performed additional simulations oranalysis. In both pictures we show lines with slope (1 − γ ) /γ for γ = 0 . , . , . , . . β nh = 1 . . γ = 0 . γ = 1 .
0. This is consistent with the scenariodescribed above, where at γ = 0 . γ = 1 . θ -point universality class. Also as predictedabove, for γ = 0 . C ( T ) T2d γ =0.1 γ =0.4 γ =1.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 C ( T ) T3d γ =0.1 γ =0.4 γ =1.0 Figure 4: The specific heat for different values of γ in two and three dimensions for n = 128.For γ = 0 . . γ = 0 . γ = 0 . θ -like transition from the swollen coil to a collapsed globule at a moderatetemperature, followed by a stronger globule-crystal transition at a lower temperature. In twodimensions, there are two peaks of roughly equal height. However, they are not well separated,which indicates that we need to go to longer lengths to study these transitions.9 .3 Low temperature phases Before considering the order of the phase transitions, especially the globule-to-crystal transition,we verify that the low-temperature phases have the properties assumed above. In particular, wedemonstrate that while the globular phase displays no orientational order, the phase for large β hb at fixed β nh is a crystal phase which displays strong orientational order by showing that inthis phase the bonds between monomers prefer to align with one axis of the lattice. ρ β hb β nh =1.0128966432 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 ρ β hb β nh =0.7128966432 Figure 5: The plots display the anisotropy parameter ρ of the system in two and three dimen-sions.To detect orientational order, we utilize an anisotropy parameter defined in Bastolla andGrassberger [10]. If we denote the number of bonds parallel to the x -, y -, and z -axes by n x , n y ,10nd n z , respectively, we define ρ = 1 . − min( n x , n y , n z )max( n x , n y , n z ) . (3.7)In a system without orientational order, this quantity tends to zero as the system size increases.A non-zero limiting value less than one of this quantity indicates weak orientational order with n min ∝ n max , while a limiting value of one indicates strong orientational order, where n max ≫ n min .We consider a fixed value of β nh such that the system is collapsed for any value of β hb . Forsmall values of β hb the polymer is in the globular phase, while for large values it is expectedto be in the crystal phase, see Figure 3. In two dimensions, we use β nh = 1 .
0, while in threedimensions we use β nh = 0 . ρ as a function of β hb for different lengths ranging from 32 to 128 in twoand three dimensions. For small β hb , we find that ρ converges to zero as n − / as expectedif only statistical fluctuations are present. Similarly, for large β hb , we find that ρ convergesto one in a corresponding fashion. This indicates the presence of strong orientational orderin the large β hb -phase, which we then deduce to be the ordered crystal. Intriguingly, for twodimensions only, there exists a value of β hb , 1 .
54, at which ρ seems to be independent of systemsize. Moreover, by choosing an appropriate exponent ( φ ≈ . As discussed elsewhere [9], when γ = 0, that is β nh = 0, there is a single phase transition whichis first-order in both two and three dimensions. We have verified that for small γ , for which wechoose γ = 0 .
1, this scenario remains intact. A bimodal distribution can be clearly seen formingas the system size becomes larger very strongly in three dimensions, and more weakly in twodimensions.To show that the θ -transition extends from γ = 1 / γ , we can focus onthe case of γ = 1, which means that β hb = 0 . hb -interactions are irrelevant.In two dimensions the maximum of the fluctuations per monomer for length n for the θ -pointbehaves as σ n ( m nh ) ∼ σ ∞ ( m nh ) + a · n φ , (3.8)where φ = 3 /
7. We have estimated from our data collected at short length that φ = 0 .
49. Thisis consistent with the observation that for finite system size the effective crossover exponentdecreases from a value well above 0 . θ -point11 ( σ ( m nh )- σ ∞ ( m nh )) / n φ ( β nh - β cnh )n φ
2d 128966432 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024-10 -5 0 5 10 15 20 σ ( m nh ) / ( n l og ( n ) . ) ( β nh - β cnh )n log(n)
3d 128966432
Figure 6: Scaling plots for the fluctuations in m nh for the θ -like transitions occuring along theline β hb = 0 . φ = 3 /
7. Having extrapolated the limiting value σ ∞ ( m nh ), we show in Figure 6 a scaling plotof the dependence of the singular part of the fluctuations as a function of β nh .In three dimensions [2, 18] theory predicts φ = 1 / β hb = 0 areshown in Figure 6. At short lengths, we find strong corrections to scaling, in accord with theobservations in [23, 24]. The effective exponent of the logarithm is equal to 2 .
8, which is aboutan order of magnitude more than the value 3 /
11 predicted [18]. However we have checked thatthe exponent decreases with the system size. 12 .4.2 Collapsed globule to folded crystal
We begin with the three-dimensional case and return to our simulations at fixed β nh = 0 . β hb we find a first-order transition from the globule to the crystal. The maximumof fluctuations per monomer σ ( m hb ) /n increases linearly in n , and the shift of the inversetemperature scales as 1 /n , in accord with finite-size scaling of a first-order transition. Figure 7shows the corresponding scaling collapse, with an extrapolated value of β chb = 1 . β nh = 1 .
0. Now, on varying β hb we find a transition which is much stronger than the θ -point, but shows no indication of beingfirst-order: the maximum of fluctuations per monomer σ ( m hb ) /n diverges with an exponentless than one. A scaling plot using a consistent power law is not convincing. Our best estimatefor the crossover exponent comes in fact from the scaling of the anisotropy parameter ρ discussedabove, which gives β chb = 1 .
54 and a crossover exponent in the vicinity of φ = 0 .
75. The scalingof the fluctuations is not inconsistent with these values. σ ( m hb ) / n ( β hb - β chb )n3d 128966432 Figure 7: A scaling plot for the fluctuations in m hb for the globule-to-crystal transition occuringalong the line β nh = 0 . We summarize our findings by presenting conjectured phase diagrams in two and three dimen-sions in Figure 8. For large values of the ratio of the interaction strength of hydrogen-bonds tonon-hydrogen bonds, a polymer will undergo a single first-order phase transition from a swollencoil at high temperatures to a folded crystalline state at low temperatures. On the other hand,for any ratio of these interaction energies less than or equal to one there is a single θ -like tran-13 nh β hb crystal swollencoil compactglobule β nh β hb swollencoil crystal compactglobule Figure 8: The schematic conjectured phase diagrams in two and three dimensions. A solidline denotes a first-order transition, and a dashed line a θ -like transition. The dot-dashed linerepresents a putative second-order phase transition that is not yet completely characterised.sition from a swollen coil to a liquid droplet-like globular phase. For intermediate ratios twotransitions can occur, so that the polymer first undergoes a θ -like transition on lowering thetemperature, followed by a second transition to the crystalline state. In three dimensions wefind that this transition is first order, while in two dimensions we find that this transition isprobably second order with a divergent specific heat. It can be argued using a zero-temperatureargument that this scenario exists as soon as the ratio of the interaction energies is greater thanone. In this way the phase diagram described is qualitatively similar to the one of the semi-stiff14nteracting polymer model described in three dimensions by Bastolla and Grassberger [10].The interesting questions that remain for future work include further characterizing theglobule-to-crystal transition in two dimensions and also clearly delineating the range of interac-tion ratios for which two transitions occur.Financial support from the Australian Research Council and the Centre of Excellence forMathematics and Statistics of Complex Systems is gratefully acknowledged by the authors. Allsimulations were performed on the computational resources of the Victorian Partnership forAdvanced Computing (VPAC). References [1] Flory P, 1971
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