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A semi-flexible attracting-segment model of three-dimensional polymer collapse

Recently it has been shown that a two-dimensional model of self-attracting polymers based on attracting segments with the addition of stiffness displays three phases: a swollen phase, a globular, liquid-like phase, and an anisotropic crystal-like phase. Here, we consider the attracting segment model in three dimensions with the addition of stiffness. While we again identify a swollen and two distinct collapsed phases, we find that both collapsed phases are anisotropic, so that there is no phase in which the polymer resembles a disordered liquid drop. Moreover all the phase transitions are first order.


Introduction
An isolated polymer in solution undergoes a collapse transition from a swollen coil to a collapsed globule as the temperature is lowered and consequently the quality of the solvent is reduced. The canonical lattice model used to describe this scenario is the model of Interacting Self-Avoiding Walks (ISAW) on a regular lattice, such as the square or simple cubic lattice [1,2] .
At high temperatures the self-avoiding walk is swollen, in that the fractal dimension of the walk d f is less than the fractal dimension of simple random walks. The exponent 1 arXiv:1411.7352v1 [cond-mat.soft] 26 Nov 2014 ν = 1/d f describes the scaling of the size of the walk, as measured for example by its radius of gyration, as a function of the length of the walk. It is known that ν = 3/4 in two dimensions [3] and ν = 0.587597 (7) in three dimensions [4].
At low temperatures the self-avoiding walk is collapsed, in that the fractal dimension of the walk d f is equal to the dimension d of the ambient space. The transition between the swollen coil and the collapsed globule happens at a particular temperature, called the θ-temperature.
In the standard description of the coil-globule transition, the transition is a tricritical point related to the N → 0 limit of the ϕ 4 -ϕ 6 O(N ) field theory [5,6,7]; there is a second-order phase transition with a specific heat exponent conjectured to be −1/3 in two dimensions [8] and 0 in three dimensions with a logarithmic divergence of the specific heat. In two dimensions the fractal dimension of the walk is expected to be d f = 7/4 [8] and d f = 2 with logarithmic corrections in three dimensions. The canonical model of interacting self-avoiding walks fits this scenario. In this model one expects the low-temperature state to be a liquid drop, i.e., the polymer is compact but disordered.
In stark contrast to this scenario, there is another simple interacting polymer model, the Interacting Hydrogen-Bond model (IHB) [9,10,11], where a pair of sites on the selfavoiding walk acquires a hydrogen-like bond potential if the sites are (non-consecutive) nearest neighbours, as in the ISAW model, and each site lies on a straight section of the walk. This model has been introduced in the context of biopolymers where hydrogen bonding plays an important role [12]. In contrast to ISAW, this model displays a firstorder collapse transition in both two and three dimensions. Here, the low-temperature state is an anisotropic compact phase described as a polymer crystal.
Another model introduced to account for hydrogen bonding is the Attracting Segments model (AS) [13,14,15] (also known as 'interacting bonds'). It is a lattice model based on self-avoiding walks where an attractive potential is assigned to bonds of the walk that lie adjacent and parallel on the lattice (though not consecutive along the walk), see Figure 1.
On the square lattice, this model seems to have two phase transitions, one of which is identified as the θ-point [15].
If one introduces stiffness into the ISAW model, one arrives at the semi-flexible ISAW model [16,17,18,19]. In addition to the nearest-neighbour site interaction of ISAW, one introduces a stiffness energy associated with consecutive parallel bonds of the walk. This was studied on the cubic lattice by Bastolla and Grassberger [16], where it was shown that depending on the energetic weighting of straight segments one finds a single first-order transition from a swollen coil to a crystalline state for a strong energetic preference for straight segments, or a soft θ-transition from a swollen coil to a liquid globule, followed by a first-order transition to a crystalline state. In two dimensions, a similar scenario has been found [20], the main difference being that the transition between the globule and the frozen state becomes second-order.
The semi-flexible AS model, in which both straight segments and interacting segments carry an energy, as shown in Figure 1, has been studied in two dimensions in [21], where it was found that it has a phase structure in common with the semi-flexible ISAW model.
In this paper, we discuss the semi-flexible AS model in three dimensions. While we again identify a swollen and two distinct collapsed phases, we find that both collapsed phases are anisotropic, so that there is no phase in which the polymer resembles a disordered liquid drop. The transitions between the swollen and each of the collapsed phases and between the two collapsed phases are first order. The three lines of transitions in parameter space meet at a triple point.

Semi-flexible Attracting Segments model
Our semi-flexible attracting segments model (semi-flexible AS model) is a self-avoiding walk on the simple cubic lattice, with self-interactions as in the AS model [13,14,15] and a stiffness (or equivalently bend energy) added. Specifically, the energy of a single chain (walk) consists of two contributions (see Figure 1): the energy −ε as for each attracting segment pair, being a pair of occupied bonds of the lattice that are adjacent and parallel on the lattice and not consecutive along the walk; and an energy −ε ss for each stiffness segment pair, being a pair of bonds consecutive along the walk that are parallel. A walk configuration ϕ n of length n has total energy where m as denotes the number of attracting segment pairs and m ss denotes the number of stiffness segment pairs. The partition function is defined then as Z n (β as , β ss ) = mas,mss C n,mas,mss e βasmas+βssmss , where β as = ε as /k B T and β ss = ε ss /k B T for temperature T and Boltzmann constant k B .
The density of states, C n,mas,mss , has been estimated by means of Monte Carlo simulations.

Simulations
On the cubic lattice we performed simulations using the FlatPERM algorithm [22], estimating the density of states up lengths for n = 128 over the two parameters m as and m ss .
We use averages for the density of states from five independent runs. Moreover, we have To detect orientational order, we estimated an anisotropy parameter [16]. Putting this broad information together it seems to indicate that there is a region in which the walk forms a swollen coil (the non-interacting SAW at β as = β ss = 0.0 lies in We thus have three clearly defined regions in the β as ,β ss -plane, delineated by rather sharp peaks in fluctuations. The presence of these fluctuations leads us to identify these regions with thermodynamic phases, so that we have a swollen globular phase, and two collapsed phases, one of which is rich in straight segments, and one of which is rich in bends. The conjectured phase diagram is shown in Figure 3. This phase diagram also indicates that the transition lines are first-order, and that the collapsed phases have in fact a crystalline structure. We will provide evidence for this in the next two sections.

Phase Transitions
In order to study the order of the three transitions, we consider the behaviour show strong transitions, and to clarify whether they are really first-order we consider the distribution of the appropriate microcanonical parameters.
In Figure 5 we show the distribution of m as at the peak of its fluctuation in β as along the lines β ss = 1.0 and β ss = −1.0, and the distribution of m ss at the peak of its fluctuation in β ss along the line β as = 1.8. The distribution along β ss = 1.0 is clearly bimodal with the distance between the double peaks slightly widening as n changes from 128 to 256. This confirms the first-order nature of that transition, and also explains why the fluctuation peak grows super-linearly in Figure 4. Along the line β ss = −1.0 we also find an emerging double peak, albeit developing at much longer lengths. Finally, along the line β as = 1.8 we find a bimodal distribution with one peak located very closely to m ss = 0. Hence, we conclude that the transitions between all three phases are first-order.

Nature of the Collapsed Phases
We now turn to the investigation of the two collapsed phases. As in [21], it will be helpful to consider an anisotropy parameter. In three dimensions, denoting the number of bonds parallel to the x-, y-, and z-axes by n x , n y , and n z , respectively, we define ρ = 1.0 − min(n x , n y , n z ) max(n x , n y , n z ) (3.1) to be the anisotropy parameter. 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 first consider the change of the anisotropy parameter as the phase boundaries are crossed along the same three lines investigated earlier. This is shown in Figure 6 for In addition to the change in m as and m ss described above, one sees that the anisotropy parameter decreases in the swollen phase, but jumps to a significantly larger value in the collapsed phases. There is a small difference in the value of the anisotropy parameter in both collapsed phases, but the change of ρ as the length of the walk increases is such that ρ might in fact tend to the same limiting value of one in the thermodynamic limit.
To test this, we now display the finite-size scaling behaviour at two points in the different collapsed phases. The values m ss /n, m as /n and ρ are shown as functions of n −2/3 for β as = 1.4 and β ss = 1.0 in Figure 9 and for β as = 1.8 and β ss = −1.0 in Figure   10. If the corrections to scaling are due to surface effects, one should find asymptotic straight lines as n −2/3 tends to zero. The limiting value of m as /n is equal to 2 in the former phase, and equal to 7/4 in the latter phase, consistent with the numerical data shown in Figure 9 and Figure 10.
In this article we have discussed a three-dimensional semi-flexible AS model. We have characterised one swollen and two collapsed phases, and transition between them. In contrast to the two dimensional version of this model, both collapsed phases are anisotropic, and none of them resemble the disordered liquid drop.