The Particle as a Biphasic Material Compound

Adsorbent materials used in chromatographic process applications are usually porous and able to swell in a fluid environment. Across the particles used in chromatographic separation, the intraparticle porosities i vary from nearly zero for pellicular stationary phases to 0.9 for low-density gels such as agarose [Carta and Jungbauer 2010, p. 70].

When the particle absorbs a small amount of solvent molecules, i.e. the liquid phase ab-sorbed by the porous resins in liquid chromatography, the aggregate can be described as a biphasic material compound. The time-dependency of a viscoelastic polymeric particle in its relation of stress and strain was already described in the previous Section 3.1.2.

The time-dependency of the deformation of this aggregate results from two concurrent molecular processes: the conformational change of the polymer network, which results in viscoelasticity, and the migration of the absorbed solvent, which results in poroelasticity [Hu and Suo 2012; Wang et al. 2014].

The theory of linear poroviscoelasticity was developed by Biot [Biot 1956, 1962, 1963]

and has since been used to study poroelastic phenomena [Nguyen et al. 2009; Hu and Suo 2012; Oyen 2014; Wang et al. 2014]. Therein, the migration of a solvent into a porous compressible material is described by the diffusion equation

∂C

∂t =D∇²C, (3.34)

whereC is the concentration field of the solvent,∇² = (_∂x^∂22 +_∂y^∂22 +_∂x^∂22)is the laplacian operator, andD is the diffusion coefficient given by [Hu et al. 2011; Wang et al. 2014]

D= 2(1−ν)Gk

(1−2ν)η . (3.35)

ν andGare the Poisson’s ratio and the shear modulus of the particle andk the intrinsic permeability of the particle. k can approximately be calculated by Darcy’s law using

k = ir²_h

K^∗ , (3.36)

where_i is the intraparticle porosity,r_h is the mean hydraulic pore radius of the particle and K^∗ is the Kozeny constant, which equals 5 for i ≥ 0.66 [Kapur et al. 1996; Lin et al. 2007].

Chapter 3. Physical Background

Equation (3.34) indicates that over a time ta disturbance diffuses over a length √ Dt. Assuming that the polymer chains of the material are incompressible and that volume change of the particle is equal to the volume of the absorbed or desorbed solvent the poroelasticity of the particle is characterized by the three parametersG, ν and D.

Hence, the viscoelastic characteristic time and the poroelastic diffusivity of the ag-gregate define an intrinsic material length scale of the agag-gregate [Wang et al. 2014].

As viscoelasticity results from molecular processes such as sliding between the polymer chains, the time of viscoelastic relaxation τ (Equation (3.25)) is independent of the length characteristic of any macroscopic observation. In contrast, poroelasticity results from the migration of an absorbed solvent, so that the time of poroelastic relaxation, as shown above, depends on the length of macroscopic observation [Hu and Suo 2012]. In compression testing, the characteristical length is often the sample diameter or during indentation the radius of the contact areaa, as indicated in Figure 3.5 [Hu et al. 2011;

Hu and Suo 2012; Hu et al. 2012; Kalcioglu et al. 2012; Wang et al. 2014].

F

Figure 3.5.: Effect of poroelasticity. A porous particle is submerged and in thermodynamic equi-librium with a surrounding solvent. By application of an external loadF by an ideally rigid inelastic compression plate the particle is pressurized locally. This causes the absorbed solvent to migrate ex-hibiting a poroelastic behavior. ais the contact radius of the particle at the compression plate.

Taking the two material specific parametersτ andD, a length √

Dτ can be calculated that represents the distance of the migration of the solvent within a time comparable to the time of viscoelastic relaxation. This length is material-specific and therefore inde-pendent of the length scale and time of any macroscopic observation [Hu and Suo 2012].

Hence, the compression behavior of a biphasic material compound can be characterized by two conditions in time. The condition t ∼ τ represents the time of viscoelastic re-laxation and the conditiont ∼L²/D represents the time and length (L) of poroelastic relaxation. As an example, when t τ and t L²/D, both processes have started and the aggregate behaves like an elastic solid with an instantaneous modulus (see e.g.

20

Chapter 3. Physical Background

Equation (3.25) for t τ) and negligible migration of the solvent. When t τ and tL²/Dviscoelastic relaxation has started, but poroelastic relaxation has already been completed. For t τ and t L²/D viscoelastic relaxation has completed, but poroe-lastic relaxation has already started. Finally, whent τ and tL²/D both processes are relaxed [Hu and Suo 2012]. A graphic illustration of various limiting conditions is given in Figure 3.6.

Figure 3.6.: Graphic representation of the limiting conditions of viscoelastic (V) and poroe-lastic (P) relaxation. Each point in the plane corresponds to a timetand a lengthL(normed to the material characteristic timeτand√

Dτ, respectively) of a macroscopic observation. The vertical line at t∼τ represents the time of viscoelastic relaxation and the inclined linet∼L²/Dthe time and length of poroelastic relaxation. For example, the point’V-relaxing, P-relaxed’ corresponds to a condition in which the migration of the solvent has already ceased, but the size of the aggregate is so small that the rate of change is limited by viscoelasticity (modified from Hu and Suo [2012]).

Based on the theory of poroelasticity, the time-dependent force-relaxation curve can be derived for a simple indentation of a poroelastic flat disk by a spherical indenter, or equivalently, for a compression of a poroelastic sphere by a flat compression plate (Figure 3.5). For a constant indentation depth or compression, the radius of the contact area,a, remains constant and the compression force decays with time. Hence, as already given above for a short-time limit, the poroelastic material behaves like an incompressible elastic material. Consequently, the force in the short-time limit is F0. In the long-time limit, when poroelastic relaxation has completed, a force F_∞ can be measured. Both

Chapter 3. Physical Background

forces are correlated by

F0

F_∞ = 2(1−ν), (3.37)

where ν is the Poisson’s ratio of the material [Biot 1941; Hu and Suo 2012; Hu et al.

2012]. The time-dependent decay of the compression forceF(t) takes the form F(t)−F_∞

F0−F_∞ =g(τ_p), (3.38)

where g is a function dependent on the type of the experiment and τp = Dt/a² is the normalized poroelastic time. For a spherical indenter of radius R pressed into a poroelastic materialF0 is given by the Hertz theory of elastic contact (Equation (3.1)) to

F0 = 16

3 Gaδ, (3.39)

where the Poisson’s ratio in the short-time limit is given byν = 0.5for an incompressible elastic material and a is given by Equation (3.5). Hu et al. [2010] found out using the finite element method thatg(τp) takes the form

g(τp) = 0.491e^{−0.908√τp}+ 0.509e^−1.679τp. (3.40)

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Chapter 3. Physical Background