parameter (maximum repulsion between two beads) can be determined from the
relationship with the Flory–Huggins parameter. The soft nature of interactions in
DPD allows to use larger time steps (this speeds up the calculations), as compared
with MD that deals with steep repulsive potentials. However, one should avoid a
too large time step that leads to numerical errors (strongly dependent on the
numerical algorithm used). Popular software packages including DPD dynamics
include e.g. GROMACS-DPD, LAMMPS, ESPResSo.
Monte Carlo (MC) methods represent a variety of models and algorithms based
on random events. MC methods often use a coarse-graining approach and a lattice
environment, instead of the continuous space, which increases the simulation speed
due to fast recognition of neighbours of a given molecule. Various methods have
been proposed in the field of simulation of conformation and dynamics of polymer
systems. The bond fluctuation model (BFM), first introduced by Carmesin and
Kremer [194] and later modified by Shaffer [195], is a good example of the MC
approach. It works on the regular cubic lattice originally proposed by Meyer for
calculation of entropy in polymer solutions [196] and next used also in the early
stage of the Flory–Huggins theory [39–44]. In this model, the excluded volume is
preserved (one element can occupy one lattice site), but the algorithm requires
usually, at least, *30% of empty lattice sites to work. The monomers can be joined
by a bond vector, which is taken from a set of 108 allowed vectors in Carmesin and
Kremer version of the algorithm. In the athermal case, a movement attempt is
realized in a set of steps: (i) a random selection of a monomer and move direction,
(ii) check if the excluded volume condition would be fulfilled and (iii) if the bond
can be created in a new position. If checks are positive, the movement of the
monomer is performed. Taking into account the temperature, additional energetic
test (using e.g. Metropolis algorithm [197]) is required, which leads to the significant extension of the calculation time.
Other classical MC methods applied for polymers are the general reptation
algorithm [198] (also empty sites are needed, but the algorithm is not ergodic), the
pivot algorithm [199] and the bond breaking algorithm [200] (working on the fully
occupied lattice, but do not preserve chains integrity). Interesting approaches called
the Cooperative Motion Algorithm (CMA) [201] and the Dynamic Lattice Liquid
(DLL) model [202] were proposed by Pakula. Both methods are based on the
cooperative movement of system elements and work on the completely occupied
lattice (no vacancies are needed). In the CMA method, system elements (solvent,
chain segments) are moved as cooperative loops starting from the temporal
vacancy, which randomly moves through the system, moving encountered elements
on the vacancy actual place until the starting point is reached. This way leads to
preservation of the excluded volume of beads. CMA is regarded as a good tool to
predict static properties and is very efficient in equilibration of polymer melts [203].
The DLL model is a dynamic method and is also based on a lattice structure; i.e.,
positions of beads (representing solvent or chain segments) are consistent with
lattice sites. It is assumed that molecular systems have some excess volume.
Therefore, molecules have enough space to vibrate around their positions (this is
considered as displacement attempts). However, moves cannot be easily performed
8 Vibrational Spectroscopy in Analysis of Stimuli-Responsive …
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relationship with the Flory–Huggins parameter. The soft nature of interactions in
DPD allows to use larger time steps (this speeds up the calculations), as compared
with MD that deals with steep repulsive potentials. However, one should avoid a
too large time step that leads to numerical errors (strongly dependent on the
numerical algorithm used). Popular software packages including DPD dynamics
include e.g. GROMACS-DPD, LAMMPS, ESPResSo.
Monte Carlo (MC) methods represent a variety of models and algorithms based
on random events. MC methods often use a coarse-graining approach and a lattice
environment, instead of the continuous space, which increases the simulation speed
due to fast recognition of neighbours of a given molecule. Various methods have
been proposed in the field of simulation of conformation and dynamics of polymer
systems. The bond fluctuation model (BFM), first introduced by Carmesin and
Kremer [194] and later modified by Shaffer [195], is a good example of the MC
approach. It works on the regular cubic lattice originally proposed by Meyer for
calculation of entropy in polymer solutions [196] and next used also in the early
stage of the Flory–Huggins theory [39–44]. In this model, the excluded volume is
preserved (one element can occupy one lattice site), but the algorithm requires
usually, at least, *30% of empty lattice sites to work. The monomers can be joined
by a bond vector, which is taken from a set of 108 allowed vectors in Carmesin and
Kremer version of the algorithm. In the athermal case, a movement attempt is
realized in a set of steps: (i) a random selection of a monomer and move direction,
(ii) check if the excluded volume condition would be fulfilled and (iii) if the bond
can be created in a new position. If checks are positive, the movement of the
monomer is performed. Taking into account the temperature, additional energetic
test (using e.g. Metropolis algorithm [197]) is required, which leads to the significant extension of the calculation time.
Other classical MC methods applied for polymers are the general reptation
algorithm [198] (also empty sites are needed, but the algorithm is not ergodic), the
pivot algorithm [199] and the bond breaking algorithm [200] (working on the fully
occupied lattice, but do not preserve chains integrity). Interesting approaches called
the Cooperative Motion Algorithm (CMA) [201] and the Dynamic Lattice Liquid
(DLL) model [202] were proposed by Pakula. Both methods are based on the
cooperative movement of system elements and work on the completely occupied
lattice (no vacancies are needed). In the CMA method, system elements (solvent,
chain segments) are moved as cooperative loops starting from the temporal
vacancy, which randomly moves through the system, moving encountered elements
on the vacancy actual place until the starting point is reached. This way leads to
preservation of the excluded volume of beads. CMA is regarded as a good tool to
predict static properties and is very efficient in equilibration of polymer melts [203].
The DLL model is a dynamic method and is also based on a lattice structure; i.e.,
positions of beads (representing solvent or chain segments) are consistent with
lattice sites. It is assumed that molecular systems have some excess volume.
Therefore, molecules have enough space to vibrate around their positions (this is
considered as displacement attempts). However, moves cannot be easily performed
8 Vibrational Spectroscopy in Analysis of Stimuli-Responsive …
249