In computer simulation of the simplest types of deformation, it assumed that the system is subjected to tensile force F (Fig.12a) or bending moment M (Fig.12b), simultaneously applied to the substrate and the polymer coating.
For polymer chains with polar groups, the parameters K2and K3can be eval-
uated from the multipolar expansion [54] of the interaction energy of two identical dipoles with electric dipole moment p placed in neighboring chains:
Fig. 10 The average normalized energy H/K1of
the element-rotator versus the number of Monte-Carlo steps NMCfor the given reduced
temperature T*= k
BT/
K1= 2. The simulation was
made on the lattices: 10 × 10 × 10 (1), 20 × 20 × 20 (2), 30 × 30 × 30 (3)
Fig. 11 The average normalized energy H/K1of
the element-rotator versus the number of Monte-Carlo steps NMCfor the given reduced
temperature T*= k
BT/
K1= 2. The simulation was
made on the lattice 20 × 20 × 20 at initial parallel (1) and chaotic (2) orientation of the rotators
K2≅ K3=
1 4πε0
⋅ p2
r3, ð7Þ
where r is the mean distance between the neighboring chains, andε0is the dielectric
constant.
Therefore, during the system deformation, i.e., for stretching and bending, parameter K2was calculated from the dependence interactions energy of chains on
the relative displacements of chain elements:
K2= K3⋅
1
r ̸r0
ð Þ3. ð8Þ
In Eq. (8), the parameter r0 is the mean distance between neighbour chain
elements in the direction of force action (along n2) at equilibrium configuration of
the system, r = r0+Δr, where Δr characterizes the change of element position
during the deformation. The energetic parameter K3keeps constant value during the
deformation.
Fig. 12 The schemes of deformation of the discrete-continuous system: the tension under the influence of the force F (a) and the bending under the influence of the moment M(b). 1—the polymer coating with the thickness h, 2—the substrate with the thickness H, 3—the position of the neutral layer, where there no tension/compression;ρ is the radius of curvature of the neutral layer corresponding to the angular deformationΔθ, O is the center of curvature
(a) Simulation algorithm
Within the framework of the simulation of the presented above system by means of the Monte-Carlo method, we generate a random process consisting of sequence of the system configurations. Using ensembles with the large enough number of configurations one can calculate average values of almost all equilibrium physical quantities. We choose the following input parameters for the simulation of defor- mations of the system shown in Fig.12: the dimensions of the “quasi-lattice” N1,
N2and N3(numbers of segments in the directions n1, n2and n3), the interaction
parameters K1and K3, the ratio of thickness of the polymer coating and the sub-
strate h/H, the length l of element-rotator, and the number of Monte-Carlo steps. All quantities having energetically dimension were normalized with respect to the parameter K1∼ 10−20 J [53], whereas length-scale parameters were normalized
using the average distance between neighbour chains of the polymer coating
a∼ 10−10m. The normalized temperature of the system is defined as T*= kBT/K1,
where kB is Boltzmann’s constant. The effective tensile stiffness ktens. and the
bending stiffness kbend.of the substrate are defined through the Young modulus of
the substrate Esubs.and geometrical parameters of the system:
ktens.= Esubs.Ssubs. L0 , kbend.= Esubs.Izsubs. L0 . ð9Þ
In (9), the quantity Ssubs.is the area of the transverse substrate cross-section in
the plane X−Z, L0is the length of the system in the non-deformable state along the
axes Y and Z, Isubs.
z =∫ x2dS is the inertia moment of the substrate cross-section with
respect to the axis Z. If the axis X and Z are the principal central inertia axes, then,
Iz= ba3/12 for the rectangular cross-section with the height a and the width
b. Since the centres of mass of the substrate and coating are not coincide wefind
that Isubs.
z = bH H
2+ 3h2
ð Þ ̸12, where H and h are correspondingly the thickness of the substrate and the polymer coating.
For the simulation of the deformations of compression/tension and bending of the system, we use the following algorithm:
• We assume the following initial state of the system. Chain elements located at the interface coating-substrate as well at ends of the system along the axis Y (in the direction n2) arefixed. At the top boundary along axis X (in the direction n1)
segments are free, and boundary conditions along the axis Z (in the direction n3)
are periodic ones since in this direction, the system is infinite. In order to achieve faster the equilibrium state at low temperatures, all segments are parallel each other, whereas for high temperatures, their orientation is given randomly. • For determination of the parameter L0in non-deformable state, we simulate its
uniform compression/extension along Y axis andfind equilibrium configuration of the system for which the energy of the polymer part (i.e., without the sub- strate), is minimal at the given temperature. For this purpose, we imply cycle of the Monte-Carlo steps with Metropolis algorithm and the self-consistency
condition. If the energy of the system decreases with the change of its linear dimensions along the Y axis, we replace the value of r0in Eq. (8) using current
value of the distance between elements in the direction n2. If the energy does not
decrease, we keep the initial value of r0.
• We simulate the tension or the plane pure bending of the system (from the equilibrium value L0). At each step of the bending deformation, we decrease the
curvature radiusρ of the neutral layer through the increase of the angle Δθ =
L0/ρ, whereas the distance n2along the axis Y between segments of the polymer
coating is determined by the formula:
r = r0+ ðH − hÞ 2 + l n1− 1 2 L0 ðN2− 1Þρ . ð10Þ
In Eq. (10), the quantities H and h are correspondingly the substrate and polymer coating thickness (Fig. 12), n1is the number of element-rotator along
the axis X. The distance between neighbour segments in Eq. (10) is defined as the distance between their geometrical centres. Since we consider thin coatings, the ratio h/H < 1, and the polymer layer at the simulation is always under tension (r > r0).
• At each step of bending deformation, we calculate the energy of equilibrium state of the system. We take the energy of elastic substrate as
Hsubs.= ktens.ð ÞΔx 2 ̸2 at the stretching deformation and Hsubs.= kbend.ð ÞΔθ 2 ̸2 at
the bending deformation.
• According of Hooke’s law, the force F applied to the system is proportional to the displacement of chain elements from their initial positions, in other words
F∼ Δx. The work of external force during the deformations transforms to the
energy change of the system: A =−ΔH, where
A =FΔx
2 . ð11Þ
Considering thatΔH = ksyst.ð ÞΔx 2 ̸2 and using the relations similar to Eq. (9)
but for whole system, we obtain that:
A =ESsyst. L0
Δx ð Þ2
2 . ð12Þ
In Eq. (12), E is the effective Young modulus of the system, the quantity Ssyst.is
the area of the transverse cross-section of the system in the plane X−Z. As a result, using Eqs. (11) and (12) it is possible to calculate, the value of F and Young’s modulus E of the system in dependence on the deformation Δx. • According to Hooke’s law the bending moment M applied to the system is
proportional to the angular displacements of the particles of the system that is
M = ksyst.Δθ, where the quantity ksyst. is the effective bending stiffness of the
curvature radius of the neutral curve Δθ = L0/ρ, we find that M ∼ 1/ρ. Note,
that the work of external loads A for an elastic system stores in the change of its potential energy, i.e., A =ΔH, where
A =MΔθ
2 . ð13Þ
Considering that ΔH = ksyst.ð ÞΔθ 2 ̸2 and using the relations similar to Eq. (9)
but for the system as whole, we obtain that:
A =EI syst. z L0 Δθ ð Þ2 2 . ð14Þ In Eq. (14), Isyst. z = b H + hð Þ
3 ̸12 is the inertia moment of the cross-section of the
system with respect to the axis Z. As a result, using Eqs. (13) and (14), we can calculate the dependencies of the moment M and Young’s modulus E of the system on the angular deformationΔθ, or on the curvature radius ρ of the neutral line of the system.
(b) The simulation results
In Figs.13,14and15, we present dependencies of the force, bending moment and
Young’s modulus depending on the type of system deformation
(stretching/bending) for different interactions of the chains and temperatures. It is seen, that with the increase of the deformations, the influence of the coating becomes more and more negligible. With the decrease of the parameter K3
describing interactions between chains (and, therefore, with the decrease of K2), the
influence of the coating disappears faster (Figs.13and14).
In Figs.13a and b, the regions placed after peaks can be explained by the break of the bonds between neighbour chains of polymer coating and the increase of a role of the substrate during the system deformation. Similar stress-strain depen- dencies have been observed during the stretching of thinfilms from.
crystalline polymers such as caoutchouc and gutta-percha [55], for which such behaviour was explained by the structural transformations on material from ordered microstructure to amorphous one. For polymers with polar groups, it is transition from ferroelectric phase to paraelectric one. Further asymptotically linear behaviour of the given curves demonstrates negligible influence of the coating and the dominant role of the substrate which deformation follows Hooke’s law.
The bending stiffness of the system consisting of the substrate and the coating (Fig.12) with different elastic properties is given as a sum of stiffness parameters of the system parts, i.e., it is given by
EIzsyst.= Esubs.Izsubs.+ Epol.Izpol.ðfor the bendingÞ. ð15bÞ
In Eqs. (15a,15b), the quantity Epol.is the elastic modulus, Spol.is the area of the
transverse cross-section in the plane X−Z, and Ipol.
z = bh 3Hð 2+ h2Þ ̸12 is the inertia
moment of the cross-section of the polymer coating with respect the axis Z. With the increase of tension deformations and decrease of interactions between chains the influence of the polymer layer on the stiffness tends to zero, i.e., Epol. → 0.
Therefore, from Eqs. (15a,15b), it follows that the relative Young’s modulus of the system“substrate-coating” E/Esubs.tends to the limit value which depends on the
ratio of the coating and substrate thicknesses h/H:
E Esubs.
→Ssubs.
Ssyst.
= 1
1 + h ̸H ðfor the tensionÞ; ð16aÞ
E Esubs. → Isubs. z Isyst.z =1 + 3 hð ̸HÞ 2 1 + hð ̸HÞ
½ 3ðfor the bendingÞ. ð16bÞ
Fig. 13 (a) The normalized force Fa/K1applied to the
system versus the relative strainΔx/L0for the given
temperature T*= k BT/ K1= 0, 1 and renormalized intrachain interaction parameter (K1= 1) and interchain one: K3= 0.1 (1), 0.05 (2), 0.01 (3); force Fa/K1
versus the strainΔx/L0for the
substrate (4) (Hooke’s law). (b) The renormalized bending moment applied to the system M/K1versus the angular
deformationΔθ for the given renormalized
temperatureT*= kBT/K1= 0,
1 and normalized intrachain interaction parameter (K1= 1) and interchain one:
K3= 0, 15 (1), 0, 10 (2), 0,
05 (3); the bending moment M/K1versus the angular
deformationΔθ for the substrate (4) (Hooke’s law)
Within this simulation, we take that the ratio h/H = 1/3, thusthe value E/
Esubs. → 0, 75 (for the tension of the system, Fig.14a), or the value E/Esubs. → 0,
5625 (for the bending of the system, Fig.14b).
Fig. 14 (a) The relative effective Young’s modulus of the system E/Esubs.versus the
relative strainΔx/L0for the
given temperature T*= k BT/ K1= 0, 1 and renormalized intrachain interaction parameter (K1= 1) and interchain one: K3= 0.4 (1),
0.7 (2), 1 (3); the limit value of the relative Young’s modulus for large deformations of system (4). (b) The relative effective Young’s modulus of the system E/Esubs.versus the
angular deformationΔθ for the given temperature T*= kBT/K1= 0, 1 and
renormalized intrachain interactions parameter (K1= 1) and interchain one:
K3= 0, 01 (1), 0, 03 (2), 0,
05 (3); the limit value of the relative Young’s modulus for large deformations of the system (4)
Fig. 15 The relative effective Young’s modulus of the system E/Esubs.versus the
angular deformationΔθ for various renormalized temperature: T*= 0 (1), 0, 5 (2), 1 (3) and renormalized intrachain interaction parameter (K1= 1) and interchain one (K3= 0, 1)
In Fig.15, the dependencies of the effective Young’s modulus of the system on the deformations level for various temperatures are shown. It is seen that with increasing temperature, the system becomes stiffer. This stiffening can be explained through the decrease of its dimensions (L0) along the axes Y and Z in the equi-
librium non-deformable state (Fig.9) and, therefore, through the increase of interactions between chains.
Varying the length of attached to the substrate chains and their density one can analyse the dependencies of the elastic parameters of the system on its dimensions along one direction or on number of polymer chains per unit surface area.