|
I. Coarse grain model for ionic and non-ionic alkylsilane self-assembly
on a hydrophilic substrate
In this
work, coarse-grained, two-dimensional models are developed for the
aqueous solutions of cationic and non-ionic alkylsilanes. The choice of
2-D models is based on the ease in modeling and ease in computations.
The cationic and non-ionic alkylsilane molecules are modeled as chains
of connected grid sites on a 2-D square lattice. Each non-ionic
alkylsilane chain consists of one hydrophilic head group and 12
hydrophobic tail groups. The cationic alkylsilanes consist of a
hydrophilic head group (non-ionic), 11 hydrophobic tail groups and a
cationic group. The cationic alkylsilane used in the experimental
studies contains a (-CH2)n segment, known as spacer, between the
hydrophilic head group and the cationic group. Given that the chemical
groups in the spacer and the hydrophobic tail are the same (-CH2), no
distinction is made between them and both are referred to as the tail
groups.
A one-dimensional model is used for the hydrophilic surface. The grid
sites (x = [1, L], y = L ) represent the surface sites on
the hydrophilic substrate in a simulation cell of size (L X L
). Under the conditions of experiments, the pH of the alkylsilane
solution is maintained at ~ 4. The result is a distribution of –OH and
negatively charged O- terminated sites on the hydrophilic surface. The
hydrophilic head groups of alkylsilane chemisorb at the –OH terminated
surface sites whereas cationic groups electrostatically physisorb to the
O- terminated sites. In the surface model used here all the surface
sites are assumed to be –OH terminated and a mean field approximation is
made for the surface charges by distributing them uniformly across the
entire surface. This choice of mean field approximation for surface
charges is similar to the Gouy-Chapman treatment of surface charges
frequently employed in the studies of surfaces. All the surface sites
are, therefore, capable of both chemisorbing the hydrophilic head groups
and interacting electrostatically with the cationic groups. In a
simulation box of size (L X L ) containing N chains
of alkylsilane, alkylsilane molecules occupy (N X 13) lattice
sites, and L sites are occupied by the surface. The remaining
empty grid sites are considered to be the water sites.
The energy model consists of nearest neighbor short-range interactions
between pairs of non-ionic groups and long-range coulomb interactions
between the pairs of charged groups. Representing water by W, head-group
by H, tail-group by T, and surface sites by S, following pair
interactions are included, εWT, εWH, εHT,
εHH, εSH, εST. The interactions between
a pair of cationic groups of two different chains, and between a
cationic group in alkylsilane and charged surface sites are described by
following Coulomb type interaction,
where Φ is the Coulomb interaction between two charged groups
separated by a unit grid distance, and d is the distance between
the charged groups. As charged groups do not see each other across the
surface, no long-range interaction is calculated between the charged
groups across the surface. The truncation method is elected for the
calculation of long-range interactions with a cut-off distance of 30
grid units. The total energy of the system is, therefore, given by the
following expression,
where εWT,
εWH, εTH, εHH, εSH, εST
are the interaction energy values between different pairs of chemical
groups, and nWT, nWH, nHT, nHH,
nSH, nST, etc. are the number of pairs of
water-tail, water-head group and so on.
All the simulations are run in a constant (N, V, T) ensemble with
periodic boundary conditions. First, N number of chains is introduced in
the simulation cell. The energy of the random configuration is
calculated from equation given above. The alkylsilane chains are then allowed to
make MC moves on the grid. Four types of MC moves have been used in the
simulations; forward reptation, backward reptation, global chain
translation, global cluster translation. A cluster is defined as a group
of chain molecules sharing nearest neighbor sites. The move is always
accepted if it results in a decrease in the overall energy of the
system. If the move results in an increases in the energy then the new
configuration is either accepted or rejected on the basis of a
transition probability, given by exp(-ΔU/kBT). Here,
ΔU is the difference between the
overall energies of new and old configurations. The evolution of
deposited films is studied using the configurational-bias Monte Carlo
(CBMC) method. The use of CBMC method is particularly important after
the adsorption of the chain molecules on the surface when it is not
possible to explore the phase space any further with the other
conventional MC moves. The simulations are run for 50-100 millions of MC
steps following the thermal equilibration.
II.
Coarse grained models for MD simulations of friction between alkylsilane
films
The alkylsilane films on the
practical MEMS structures are typically deposited on an amorphous silica
substrate. However, in MD simulations it is convenient to model
alkylsilane films bonded to a crystalline substrate while keeping the
surface density of chain molecules similar to what is seen in the
experimental films. In the current work, an α-cristobalite structure is
used to model the silica substrate. The unit cell for this structure has
dimensions of 4.97 X 4.97 X 6.92 Å3. The silica surface for
monolayer deposition is constructed from an (10 X 10 X 2) array of the
unit cells of cristobalite, leading to surface dimensions of 4.97 X 4.97
Å2 in the simulation cell. The silica substrate in this work is
considered to be rigid, i.e., the real dynamics of the substrate atoms
is not simulated. Therefore, effectively, this substrate structure
serves the purpose of providing the sites for alkylsilane chain
attachment and as a means to slide the films against each other.
Figure 1. A representation of α-cristobalite substrate. The red atoms
represent O and the yellow atoms represent Si.
The alkylsilane films in the experimental MEMS structures are
chemisorbed to the Si atoms on the silica substrate via a Si-O-Si
linkage (Fig. 5.2). Also, the head groups (Si(OH)3) of the alkylsilane
chains are known to crosslink, leading to a very rigid alkylsilane-substrate
interface structure. Because of this rigid interface, it is convenient
not to simulate the real dynamics of these (Si(OH)3) head groups.
Therefore, a simplified model is used with the hydrocarbon tail of
alkylsilanes attached to the terminal Si groups on the cristobalite
substrate. The united atom model is used to represent CH2 and CH3 groups
as a single entity in the hydrophobic tail of the alkylsilane. Such
coarse graining of the complex molecules is widely accepted in the MD
simulations and describes the properties accurately when coupled with
appropriate potential models. The structure of the model substrate
provides one site per unit cell for the alkylsilane chains. Therefore,
the simulation cell consists of 100 alkylsilane chains on each
substrate. This structure results in a density of 24.7 Å2 per chain that
is consistent with the density of alkylsilane films seen in experiments
on amorphous silica substrates.
Figure 2. A simplified representation of experimental alkylsilane films
on a silica substrate.
In the model developed here, the alkyl chains are attached to the
terminal Si atoms on the α-cristobalite surface keeping the bond angles
and bond lengths as specified in the universal force field (UFF). A
mirror image of the bottom substrate and film is then used to generate
the top substrate and film at a given distance from the bottom film. A
schematic representation of the two alkylsilane coated α-cristobalite
substrates is shown in Fig. 3. The substrates are made infinite in the X
and Y directions by employing two-dimensional periodic boundary
conditions. To simulate the friction between films, the top and bottom
rigid substrates are set in motion at a constant velocity in the
opposite directions. The structure of alkylsilane films then evolves
following Newton’s equations of motion.
Figure 3. A schematic representation of the initial structure of the
alkylsilane films on -cristobalite substrates used in the friction
simulations.
A molecular mechanics potential model is used to describe the
interactions between different species in the simulation cell. The
potential model includes intra-chain bond stretching and bond bending
potentials, and also inter-chain and intra-chain 12-6 LJ potential. The
intra-chain LJ potential is used for groups that are separated by three
or more bonds. The interactions between the alkyl tails and the silica
substrate are modeled as 12-6 LJ potential. The total potential energy
of the system is given by the equation (2.28). The interaction
parameters for the various potential terms are taken from a Universal
Force Field (UFF).
The frictional behavior of alkylsilane films in this work is
characterized in terms of the friction coefficient of these films at
different conditions. The friction coefficient in macroscopic
experiments is defined as the ratio of the forces in the sliding
direction Fx, and in the direction normal to sliding Fz, and is
expressed as,
In the MD simulations performed here, the sliding and normal forces on
the film are computed as the sum of the respective force components on
atoms comprising a film and its corresponding substrate. The
instantaneous force values are recorded during the course of the
simulation, and the friction coefficients are determined as the ratio of
the time averages of the sliding force and normal force,
The time averages of the sliding, and normal forces are computed by
taking block averages of the force values, and the standard deviation of
the average force values is calculated as the standard deviations of the
block averages.
|
