In the present work, we compare different representation models of the polymer star. We use three models: coarse-grained (CG), mean-field (MF) and the hybrid of mean-field and coarse-grained representations; illustrations of these models are presented in Fig.4.1. The polymer star consists of f arms (we limit to f = 3 value) withNsegments per arm, the total number of segments could be counted as f N.
Coarse-grained [3D] Hybrid [3D] Mean-field [1D]
explicit segments
volume density
volume density
explicit segments
FIGURE4.1: Schematic representation of different models: (from left to right) the coarse-grained model in 3D, the hybrid model in 3D and the mean-field model in 1D. The coarse-grained model consists of ex- plicit segments only (red beads). The hybrid model consists of both explicit segments (red beads) and implicit segments (blur). The mean-
field model consists of implicit segments only (blur).
4.3.1 Coarse-grained model
In simulations, we embrace the coarse-grained (CG) model for computational rea- sons. The description was given earlier (see Chapter3).
4.3.2 Mean-field model
The description was given earlier (see Chapter3).
4.3.3 Hybrid model
The hybrid MC-SCF method is based on a combination of the mean-field and the coarse-grained models. Instead of MD, we use the MC solver for the coarse-grained model. It represents selected segments of the system as explicit particles, and the remaining (implicit) segments as a density field, shown in Fig.4.1. All work now is done in a 3-dimensional box of lattice sites, each coordinatex-y-zhasMlattice sites, whereMis large enough so that the system boundaries do not affect the conforma- tional properties of the central star.
Idea of MC-SCF fragments
With a protocol discussed below the star, the molecule is split into a number of frag- ments, which we may numberk = 1, . . . ,K. Each of these fragments has a lengthn segments and is bracketed by explicit segments. The task for SF-SCF is to compute for each of these fragments the Helmholtz energyFk (as well as the distributions of
the segments). For this, we slightly deviate from the above SF-SCF protocol. Instead of a one-gradient approach, we now have to consider three gradients. The first and last coordinates (explicit segments) of a given fragment numberkare given by⃗rk0and r
⃗kn+1, respectively. These coordinates are provided by the MC protocol. We design a sub-box around these two points so that with SF-SCF we can efficiently compute the Helmholtz energy for this sub-box. The size of the sub-box ism×m×mlattice sites, where we notice that frequentlymmay be significantly smaller thanM. The position of the boundaries of the sub-boxeskis ideally far away from the coordinates of the two explicit segments so that the fragment in between the constraining segment can not reach the sub-box boundary. Nevertheless, at the boundaries of the sub-boxes, we implement reflecting (mirror-like) boundary conditions so that in cases that the sub-walks hit the sub-box boundaries the adverse effects are minimized.
The MC-SCF hybrid is rather flexible in how the workflow is distributed be- tween the MC and the SCF parts. When computational efficiency is important one typically should aim for long fragmentsnand thus few MC-degrees of freedom (rel- atively few explicit segments). In this case, the MC-sampling is less demanding and this outweighs the increase in workload for the SCF part. Inversely when the “cor- rectness” of results is most evident, one should strive for more MC particles (explicit segments) and thus shorter fragments. The more explicit segments in the model the higher is the potential to account for the excluded volume effects.
SCF procedure in the MC-SCF method
We have to extend the above protocol so that the SCF equations take three spatial coordinates (r⃗ = (x,y,z)) into account. So both the potentials and the densities are now needed for all spatial coordinates. Now L(⃗) =r 1 for all coordinates. Let us assume that for the sub-boxk we know the potentials for all its coordinates. The evaluation of the polymer volume fractions for fragment k requires the use of the composition law, which now features two end-point distributionsGk(⃗r,s|⃗rk0, 0)and Gk(⃗r,s|⃗rkn+1,n+1), as both the begin as well as the end of the walk of the fragment is specified:
φk(⃗) =r
∑
n s=1CGk(⃗r,s|⃗rk0, 0)Gk(⃗r,s|⃗rkn+1,n+1)
G1(⃗)r , (4.1)
where againCis computed such thatn=∑
r
⃗
φk(⃗)r . The propagators initiation can be done as the followingGk(⃗r, 0|⃗rk0, 0) =1 for⃗r=⃗rk0and zero otherwise, and similarly, Gk(⃗r,n+1|⃗rkn+1,n+1) = 1 when⃗r =⃗rkn+1 and zero otherwise. The propagators now read
Gk(⃗r,s|⃗rk, 0) =⟨Gk(⃗r,s−1|⃗rk, 0)⟩G1(⃗r), (4.2) Gk(⃗r,s|⃗rk,n+1) =⟨Gk(⃗r,s+1|⃗rk,n+1)⟩G1(⃗r). (4.3) In the three-gradient system, the angular brackets are computed as
⟨G(x,y,z)⟩= 1
6(G(x−1,y,z) +G(x+1,y,z) +G(x,y−1,z)+
G(x,y+1,z) +G(x,y,z−1) +G(x,y,z+1)). (4.4)
The overall volume fraction of the segments that are accounted for by SCF is found by a summation over all the sub-boxes:
φ(⃗r) =
∑
k
φk(⃗)r . (4.5)
The corresponding potentials are found by Eq.3.25implemented for thex-y-zcoor- dinates and with the proper definition of the angular brackets. The evaluation of the solvent density and potentials are computed likewise.
With this revised protocol it is possible to find again the SF-SCF fixed point. The overall Helmholtz energy (a 3-gradient variant of the Helmholtz energy is obtained similarly as in Eq.3.28with straightforward minor adjustments) for the overall sys- tem is found by the summation over the SF-SCF Helmholtz energies per sub-box, that isF =∑
k
Fk. This overall Helmholtz energy is to be used in the MC protocol.
MC procedure in the MC-SCF method
There are a few salient features that we need to mention at this stage. Typically, the explicit segments are assumed to have a density of unit at the specified coordinates and therefore the implicit segments cannot enter the site already taken by the explicit segments. This is implemented by putting the statistical weightG1to zero for these
“taken” sites. When the sub-boxes overlap it can happen that in the sub-box explicit segments occur that are linked to other fragments. Then also for these taken sites the statistical weightsG1 are set to zero. Hence, all coordinates that are occupied by explicit segments were excluded for the implicit ones. Secondly, by virtue of the cubic lattice, starting and finishing positions for a fragment with lengthn are allowed. There is an even/odd problem, but this problem is trivially accounted for in the MC protocol that generates the starting and stopping coordinates for each fragmentk.
The task for the MC protocol is to find successive explicit particle positions in order to sample the positional degree of freedom of the star’s segments. The move- ment is driven by Monte Carlo (MC) protocol. We implemented Metropolis–Hastings algorithm with nested Monte Carlo cycle with an approximate inner potential within [83]. In the inner loop the acceptance criterion for the MC trial moves is the standard Metropolis criterion [84], but instead of the internal energy of the system, the crite- rion uses the potentials of mean force (as computed by a Helmholtz energy equiva- lent to Eq.3.28, but of course computed for the model used in the hybrid strategy as already mentioned)F, as an input.
The probability of acceptance for MC moves is the following:
p=min(︂1, exp{−β[Fnew(⃗r′)−Fold(⃗)]}r )︂, (4.6) where Fnew and Fold are the potentials of mean force with the explicit segments placed at coordinates⃗r′ in the old configuration and at⃗r in the new configuration, respectively.
In the outer loop we use Metropolis-Hastings acceptance criterion:
p=min (︄
1,πjπ′i πiπ′j
)︄
, (4.7)
where πj and πi are probabilities of trial and current configurations, respectively, calculating by the SCF part of the MC-SCF method. Theπ′j and π′i are probabili- ties of trial and current configurations calculating (in the inner loop) by following approximate potential:
Fk = 3 2
R2k N +Vk
(︄
(1− N Vk)ln
(︄
1− N Vk
)︄
+χN
Vk(1− N Vk)
)︄
, (4.8)
where Vk = R3kN3ν, for all solvent quality conditions ν = 0.9. The approximate potential consists of 3 terms: conformational, translational entropy of solvent, and interactions term.
To summarize the above hybrid protocol we can say that the potentials of mean force correspond to the Helmholtz energies obtained from the SF-SCF calculation keeping the explicit segments fixed at given positions and averaging over all possi- ble locations of the remaining implicit segments [40,72]. In this way, we can account for local density fluctuations on the level of explicit segments, beyond the mean-field level, and simultaneously reduce the computational cost as compared to explicit par- ticle simulations.
MC trials
We have implemented three types of MC-moves: pivot, “one-bond”, and one-node movements. We randomly choose a type of movement on each MC step. By this, we try to sample configurational space as fast as possible.
The idea of the pivot movement is shown in Fig.4.2. Firstly, we randomly choose the explicit node in one of the star arms (the empty segment in Fig.4.2). After that, starting from the chosen node and further to the end of the star arm, we rotate all nodes as one rigid object with an angleθ. The angle of rotation due to simple cubic geometry could be only±π/2.
The “one-bond” movement is another type of movement in our implementation.
This type of movement is needed because the pivot movements cannot reach all degrees of freedom. The “one-bond” movements fill in for these missed positional degrees of freedom. By this movement only one “bond” in the system changes, by “bond” now we mean implicit segments between two neighboring nodes. We choose randomly one node and therefore we split the star into 2 sets of nodes. The first set is all nodes excluded the chosen node and nodes, which go after the chosen one. The second set of nodes is the total number of nodes minus the first set. By using these two sets we can move only one “bond” of the star. This means that all nodes in one of the sets are shifted by the same amount.
The one-node move is the simplest movement in the MC-SCF method. By this movement only one randomly picked node is moved by one lattice point.
Using these three types of movements we successfully sample polymer star at any solvent quality conditions using the MC-SCF method.
4.3.4 Simulation protocols
We perform simulations for a number of solvent qualities for one polymer star with f = 3 arms and a different number of segments per arm Nin a range fromN = 20 up toN =200.
We choose appropriate box sizes for each of the methods. The box size at least must fit the star and should exceedN. In principle, the upper limit is not restricted.
FIGURE4.2: The idea of the pivot movement. Circles represent nodes, blue connections are implicit polymer segments.
Simulation boxes for the MD and the MC-SCF methods were identical. The perfor- mance of an MD simulation strongly depends on the number of beads (segments) and not on the box size. In the MC-SCF method, the performance does depend on the size of the box size. More accurately, it depends on the sizes of the MC-SCF frag- ment (sub-box) and the number of fragments in the star. Taking this into account, we varied box size fromboxl = 32 forN = 20 up toboxl = 90 forN = 200. In the 1D- SCF method, the simulation box was chosen in order to satisfyboxl = N[units] +5 so that none of the segments of the star can reach the system boundary.
For the MD method we equilibrated all systems as long as necessary, after that we run production simulation for 24 hours. For the MC-SCF method we run simulation only for 24 hours. All methods we simulated on similar, in terms of efficiency, CPU cores. For comparison reasons, we used only one CPU core from one processor per simulation. For small systems (N ≤ 60) this condition was more than sufficient to reach good statistical accuracies. For the 1D-SCF method, all these restrictions are irrelevant because the calculation of any of the systems took less than 1 CPU minute.
MD method
The simulation protocol for the MD method is straightforward. We construct an initial configuration of the star. After construction, we turn on the bonded and non- bonded interactions between all segments. After that, we do a sufficiently long equi- libration run. Finally, we let the system evolve while data is collected.
1D-SCF method
In order to simulate polymer star using the 1D-SCF method, we apply spherical symmetry. We consider only one linear chain fixed to the center of the simulation box with a certain volume density, which corresponds to the defined number of the arms Fig.4.3. After SF-SCF minimization, we have a polymer star in an equilibrium state for a specified value of theχparameter.
MC-SCF method
A simulation protocol for the MC-SCF method can be summarized as follow:
1. construct the initial configuration of the system;
2. calculate the free energy of the systemFoldusing the SCF part of the MC-SCF method and approximate potential Eq.4.8;
inner loop:
for each inner step (we do 3 times) do following:
FIGURE 4.3: Schematic representation of 3-arm star in mean-field model in 1D-SCF.
(a) choose an explicit segment;
(b) choose a type of movement (pivot, “one-bond” or one-node);
(c) move corresponding (to movement) explicit segments;
(d) calculate the free energy of the systemFnewusing Eq.4.8;
(e) accept with probabilityP(see Eq.4.6) and update the position of explicit segments. When the new positions are rejected, return all explicit seg- ments to old positions;
3. calculate the free energy of the system Fnew using SCF part of the MC-SCF method and approximate potential Eq.4.8;
4. accept with probabilityP(see Eq.4.7) and update the position of explicit seg- ments. When the new positions are rejected, return all explicit segments to old positions;
5. go to inner loop.
We emphasize that the MC-SCF method distinguishes all explicit segments of the star. By choosing one explicit segment we choose (in pivot move) only one arm of the star.