The characteristics mentioned above were probed using the aforementioned methods of Step Shear, Os-cillatory Rheology and Microrheology.
1.2.1 Step Shear
The stress-strain curve of the material can be obtained by application of a shear stress, the stress applied makes the material follow a shearing flow which in this case resembles the one obtained by the plane Couette geometry [24].
The plane Couette geometry creates a shearing flow by containing the fluid between two parallel plates, then one of the plates slides in a certain direction and generates a flow created by the shear stress that is being exerted onto the material. Computationally what is done is deforming the simulation box step by step at a certain rate [14].
Initially the simulation box is defined by three vectorsA⃗ = (α,0,0),B⃗ = (0, β,0)andC⃗ = (0,0, ζ) (Fig. 1.5 left), in this specific case the magnitude ofα,β andζis the same which makes the simulation box a cube, in order to allow for shear to happen the simulation box is shifted from orthogonal to triclinic and the vectors that define the new box will be given by⃗a = (α,0,0)⃗b = (δ, β,0)and⃗c = (0,0, ζ) (Fig. 1.5 right). The tilt factorδis increased at everyN steps, as this value changes one of thexzplanes of the simulation box will slide towards the positive xaxis leaving the otherxzplane unchanged as it would happen in the two sliding plates case.
x y
z
A⃗ B⃗
C⃗
x y
z
⃗a
⃗b
⃗c
Figure 1.5: Cubic simulation box with vectorsA, ⃗⃗ B, ⃗C (at the left), and triclinic simulation box with vectors⃗a,⃗b, ⃗c(at the right).
The structure suffers a transformation which can be defined as the following shear matrixTin which the new coordinatesr1of the particles will be given after the first transformation by
r1 =T r0, (1.5)
in whichT can be given by
T =
1 δT 0 0 1 0 0 0 1
, (1.6)
this transformation is applied multiple times. The time between transformations is given by N, the number of timesteps for which the system relaxes. Increasing or decreasing this number will change the
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rate at which the shear transformation is applied, a higher N will mean a slower rate. If the integration timestep isτi, the shear rateγ˙ = ∂γ∂t can be calculated as
˙ γ = ∂γ
∂t = ∂γ N τi
. (1.7)
These simulations are done at zero temperature (athermal) in order to neglect effects from the random term of the Langevin dynamics, as this affects the kinetic energy.
The computation of a stress tensor is then done at each deformation by inspecting the interactions of every particle, this tensor has six different terms regarding the surface and direction at which stress is being exerted. It is calculated in the following way as described by Thompson et al. [27],
σij = PN
k rkifkj
V , (1.8)
the sum is done over allNparticles,i, jcan bex, y, z;rkiis particlek’s position regarding componenti andfkjis particlek’s force over componentj. In this case the kinetic part of the stress tensor is neglected as the velocities of the particles are low.
When performing these measurements one should also be careful with the boundary conditions, initially Periodic Boundary Conditions [28] are used. This means that particles leaving the box will be added back on the opposite plane where they left. To every coordinate the PBC function described on Eq. 1.9 is applied wherelis the coordinate andLthe length of the box,
PBC(l) =
l+L ifl <0
l−L ifl > L . (1.9)
This is equivalent to having 26 simulations boxes around the main simulation box, so that particles in its limits are also interacting with neighbouring particles that might be in the opposite part of the box.
In Fig. 1.6 a representation of this idea is made in two dimensions, particles near the edge of the box interact with particles on the opposite edge and, if they cross the limit of the box they will re-appear on the opposite edge. This periodicity in space is what characterizes and defines this type of boundary conditions, particles always interact with each other over the closest distance possible.
Figure 1.6: Representation of what happens in Periodic Boundary Conditions in two dimensions, particles always interact with particles close to them even if they’re far from each other in the simulation box. Their real distance is measured as if there were other simulation boxes around them.
When shear stress is applied, the simulation box becomes triclinic, periodic boundary conditions can 12
no longer be considered. This is due to the particles that might leave the box in one of the planes that is suffering shear stress, or in the opposite plane, these particles are added to the box, as they normally would, with an offset equivalent to the tilt factorδ. These are Lees-Edwards boundary conditions [29], a graphical example of both aforementioned examples is shown in Fig. 1.7.
Figure 1.7: In this image the two types of boundary conditions are represented. On the right, Periodic Boundary Conditions are shown, the particle leaving the box in the inferior limit, rejoins the box in the upper limit following Eq. 1.9. On the left, Lees-Edwards conditions are shown, the particles are added back in the box like in the previous case but if they leave in a surface that is suffering shear (or the opposite plane), they are added back with a shift.
1.2.2 Oscillatory Rheology
A method to study the characteristics of the material at different timescales is also defined by imposing a periodic strain of small amplitude. This method will not tell us the lengths at which the system is able to deform and its internal stress but how this internal stress varies at each frequency of oscillation. From this definition the computation of the viscoelasticity coefficientsG′andG′′can be done, these values, as said before, measure the elastic and viscous response of the material.
In practice the shear deformation is done repetitively at a low strain amplitude in a periodic manner, the valueδwill vary with time with the following equation:
δ=γ0 sin(ωt), (1.10)
in which Ais the amplitude of oscillation and ω the frequency of oscillation. The amplitude must be chosen with close attention to the range within which the behaviour of the material is linear. We can see this by inspecting the stress-strain curve obtained by the step shear analysis.
As before we measure the stress tensor and use the computed values to obtain the viscoelasticity coefficients in the following way [14].
G′1(ω, γ0) = ω γ0π
Z t0+2π/ω t0
σxy(t) sin(ωt)dt , (1.11)
G′′1(ω, γ0) = ω γ0π
Z t0+2π/ω t0
σxy(t) cos(ωt)dt , (1.12) The values of the stress tensor are integrated starting from an initial random timet0 until the end of the oscillation att0+ 2π/ω, contemplating the whole period. We obtainG′1andG′′1which are the first order viscoelastic coefficients, these represent only the response of the material in the linear regime which is the region that is being studied here. This definition comes from the Fourrier series that represents stress σ,
σ(t;ω, γ0) =γ0
X
n odd
G′n(ω, γ0) sin(nωt) +G′′n(ω, γ0) cos(nωt) , (1.13)
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linear regime contributions will be given by the first harmonicG′1andG′′1
1.2.3 Microrheology
Microrheology covers another range of methods for analysing the material’s rheology as the ones men-tioned before. In these methods however there’s not a physical perturbation of the material as the ap-plication of a shear force, simple observation is done, and unperturbed particles are studied only subject to their own Brownian motion [22]. This is also called passive Microrheology as opposed to the active one previously defined. Passive microrheology usually provides more detailed measurements which are not affected by external factors. Additionally, it provides a broad range of oscillation frequencies, much wider than oscillatory rheology does.
Mean-Squared Displacement
Firstly, the Mean-Squared Displacement (MSD) is used to characterize the material’s viscoelasticity [30]
this is done by using the Generalized Stokes-Einstein Relation (GSER) [31]:
G∗(ω) = kBT
iω⟨∆r(ω)2⟩πa, (1.14)
whereG∗(ω) is the viscoelasticity coefficient (a complex number),⟨∆r(ω)2⟩the Fourier transform of the MSD andathe radius of the particle we’re considering.
The MSD is calculated at each timestep as an ensemble over all particles as stated in equation 1.15,
⟨∆r(τ)2⟩=⟨(r(t+τ)−r(t))2⟩. (1.15) This MSD is not useful in the time-space but rather in frequency, however to preform a Fourier transform data from 0 to∞ would be needed. As this is not possible different strategies need to be applied to achieve⟨∆r(ω)2⟩
The method used by Mason [31] is here applied, firstly the logarithmic derivativeαis found at every timestep,
α(τ) = d ln⟨∆r2(τ)⟩
d lnτ , (1.16)
in practice this is done by checking the slope of every point in the logarithmic space.
The module of the viscoelasticity coefficient is given then by the following expression,
|G∗(ω)|= kBT
πa⟨∆r2(τ = 1/ω)⟩Γ [1 +α(τ = 1/ω)], (1.17) whereΓis the gamma function, this is an approximation of the GSER.
The real and imaginary part of this viscoelasticity coefficient is then retrieved in the following way:
µ(ω) = π 2
d ln|G∗(ω)|
d lnω , G′(ω) =|G∗(ω)|cos (µ(ω)), G′′(ω) =|G∗(ω)|sin (µ(ω)), (1.18) whereG′(ω)represents the elastic response of the material andG′′(ω)the viscous response.
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Two-Point Microrheology
This method is fairly similar to the previous one but in this case the motion of pairs of particles is used to compute not an MSD but a two point MSD which will be the cross correlated motion of that pair of particles. The advantages of this method are its robustness to uncertainties and its ability to probe the bulk rheology of a material in a more exact manner, it also takes away effects of the whole gel’s movement [22]. If∆r1(τ)is the displacement of particle 1 regarding its initial position and∆r2(τ)the one of particle 2 their correlation can be represented as
Drr(R, τ) =⟨∆r1(τ) ∆r2(τ)⟩, (1.19) in which R is the module of vectorR⃗ that connects the two particles, the displacements∆r1(τ) and
∆r2(τ)are calculated in the line that connects the centres of the two particles 1 and 2, so∆r1(τ)and
∆r2(τ)will be the actual displacements of the particles (∆rreal1 and∆rreal2 ) projected on vectorR⃗ (Fig.
1.8).
P1 R⃗ P2
∆rreal1 (τ) ∆rreal2 (τ)
∆r1(τ) ∆r2(τ)
Figure 1.8: Representation of the displacement for two point Microrheology. The displacement is measured in the line that connects both centres of the particles to reduce noise. The real displacement of the particles is represented, as well as the projection of this displacement on the line that connects both centres.
The cross correlated motions are aggregated depending on the distance between the two particles at an initial step. In practice what is done is the binning of the particles depending on their initial distance so that we have a certain number of points inR. Drrwill be then a two-dimensional array in space and time.
This array is transformed in the two point MSD by getting rid of the spatial dependence and re-scaling it with distance as:
⟨∆r2(τ)⟩2= 2R
a Drr. (1.20)
Then the same procedure as before is used to transform this two-point MSD into the viscoelasticity coefficient.
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Chapter 2
Computational Methods
On this chapter an overview of the computational methods used is given. Molecular Dynamics [28] was mainly used to perform the simulation of the gel and in addition to this a percolation analysis algorithm, Hoshen-Kopelman [32], was also developed. In the next sections the parameters and details of the Molecular Dynamics simulation are explained as well as the tools used to do it.