The Viscoelastic Particle

Many resins used in chromatographic separation at preparative scale are of polymeric nature and known to exhibit a time-dependent behavior in their relation of stress and strain. Figure 3.3 shows this relation of the strainε(t)of a material under the application

14

Chapter 3. Physical Background

of a stressσ(t)during a time period t1−t0. The strain typically shows an initial elastic response (OA) to the applied stress followed by delayed elastic strain (AC). In the case of creep, the material additionally will acquire a steadily increasing creep strain (BC). By removing the stress, there is an immediate elastic response (CD) followed by a delayed elastic response (DE) [Johnson 1985, p. 184-185].

t σ, ε

t0 t1

σ(t)

ε(t)

O A B

C D

E

Figure 3.3.: Viscoelastic material behavior in the relationship of stress and strain. The material exhibits a time delayed response ε(t) on a stress function σ(t). The strain shows an initial elastic response (OA) to the stress followed by delayed elastic strain (AC). In the case of a creeping material, a steadily increasing creep strain (BC) can be observed. Removing the stress, an immediate elastic response (CD) followed by a delayed elastic response (DE) is exhibited (modified from Johnson [1985]).

As a result of the action of creep, a permanent strain is left after total relaxation of the material (E). The viscoelastic behavior, having elastic and viscous components, can be treated as a linear stress-strain relationship. This requires that the strains are sufficiently small (as in the Hertz theory of elasticity) and Boltzmann’s superposition principle is valid [Johnson 1985, p. 184-185]. Thus, for linearity, an increase in stress by a constant factor causes an increase in strain by the same factor. According to the superposition principle, the strain response to different stress histories acting simultaneously must be identical to the sum of strain responses to the stress histories acting separately [Johnson 1985, p. 184-185]. Consequently, the viscoelastic material behavior is often expressed as linear combinations of springs and dashpots. Different models exists covering various combinations and arrangements of these elements such as the well-known Maxwell or Kelvin-Voigtmodel. Therefore, in the former model, spring and dashpot are in series and in the latter in parallel. Nevertheless, these simple models are often proved insufficient as the Maxwell model does not account for creep or recovery and the Kelvin-Voigt model

Chapter 3. Physical Background

does not describe stress-relaxation. Thestandard linear solid model (also known as the Zener model [Zener 1948]) or the Prony series model consider both phenomena. Both models are described in detail in the following sections.

The Standard Linear Solid (SLS) Model

Two types of the Standard Linear Solid (SLS) model are frequently used: The Kelvin type [e.g. Cheng et al. 2005; Toohey et al. 2016] and the Maxwell type [e.g. Nguyen et al.

2009; Tirella et al. 2014; Mattei et al. 2015]. The former is composed of a Voigt element in series with a free spring (Figure 3.4 A) and the latter model consists of a Maxwell element in parallel with an equilibrium spring (Figure 3.4 B). Both types accurately describe the time-dependent behavior in the relation of stress and strain of a viscoelastic material.

Figure 3.4.: Schematic representation of the SLS model types. (A) shows the Kelvin type of the SLS model as a Voigt-element (spring and dashpot in parallel) in series with a free spring and (B)shows the Maxwell type of the SLS model as a Maxwell-element (spring and dashpot in series) in parallel with an equilibrium spring. G0 is the shear modulus of the free spring and G1 and η are the shear modulus and shear viscosity of the spring and dashpot of the Voigt or Maxwell element, respectively. εandε˙ are the strain and strain rates of the springs and dashpots.

The force-displacement relation F(δ) during the compression of a viscoelastic sphere (radius R1) by an ideally rigid incompressible flat plate (radius R2 → ∞) can be de-scribed in analogy to the Hertz elastic theory [Johnson 1985, p. 184-195; Yan et al.

2009]. Assuming a time-independent Poisson’s ratio, the time-dependent load based on a rate-controlled displacement, δ(t), is given in analogy to Equation (3.1) as [Lee and Radok 1960; Johnson 1985, p. 184-195; Mattice et al. 2006; Yan et al. 2009; Mattei et al. 2015; Toohey et al. 2016]

F(t) = 4

Chapter 3. Physical Background

where the equivalent radius in this case is given by1/R^∗ = 1/R1+ 1/∞ ≈1/R1 = 1/R. G(t) is the time-dependent shear modulus of the particle which is correlated to the Young’s modulus E(t) by

E(t) = 2G(t)(1 +ν). (3.24)

Equation (3.23) exactly describes the Boltzmann postulation that a force incrementdF through time t depends on an event happened in the past, which is the increment of strain d(δ^3/2(t^∗)) at the time t^∗, as well as on the stress-relaxation over the time span (t−t^∗). The overall shear modulus of the three parameter SLS model can be derived, e.g for the Maxwell type, as (see Appendix C.2 for details)

GSLS(t) = G0+G1e^−t/τ. (3.25) where G0 and G1 are the shear moduli of the spring and dashpot, respectively, and τ =η/G1 is the characteristic relaxation time of the viscoelastic material. The integral in Equation (3.23) is evaluated using Laplacian transformation (analog to Lee and Radok [1960] and Toohey et al. [2016]; see Appendix C.3 for details) and leads to

F(t) = wherev = ∆(t)/t is the displacement velocity of the indenter and erfi is the imaginary error function.

The Force-Relaxation Model

The compression load relaxation answer of a viscoelastic material can be described by a force-relaxation model expressed as Prony series. The modelling framework was pre-sented by Mattice et al. [2006] considering indentation by a spherical probe. This ap-proach was adapted successfully later by Yan et al. [2009] for the compression of spherical agarose micro-particles. Thus the constant2G(t), in Equation (3.24) can be replaced by the following relaxation functionG(t) represented by Prony series [Mattice et al. 2006]

G(t) =C0+

N

X

i=1

Cie^−t/τi. (3.27)

Chapter 3. Physical Background

Then, the load relaxation solution has the form

F(t) =B0+

N

X

i=1

Bie^−t/τi. (3.28)

Here, N is the number of force relaxation functions contributing to the overall time-dependent force response. The relaxation time τ_i characterizes the mean time span in which a quantity exponentially decayed to 1/e = 0.368. The fit coefficients B₀ and B_i are related to the relaxation coefficientsC0 and Ci by

C0 = B0

where RCF is the ramp correction factor considering the creep that occurs during the ramp loading period of τR [Mattice et al. 2006; Oyen 2014; Toohey et al. 2016]

RCFi = τi

From the obtained relaxation coefficients, the instantaneous shear modulus G(t = 0) and therelaxed modulus G(t → ∞)can be calculated by

G(t= 0) =

Instantaneous and relaxed Young’s modulusE(t= 0) and E(t→ ∞) are then given by Equation (3.24) indicating the apparent stiffness of the particle at the onset of compres-sion and at completely relaxed state, respectively.

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Chapter 3. Physical Background