When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces, which are hard to quantify experimentally, in particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure quantifies relevant physical parameter such as the Laplace pressure, Darcy's permeability, effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing into the pore. In determining all these parameters independently, the consistency of our analysis of such forced imbibition processes is demonstrated.
Deep Dive into Forced Imbibition - a Tool for Determining Laplace Pressure, Drag Force and Slip Length in Capillary Filling Experiments.
When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces, which are hard to quantify experimentally, in particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure quantifies relevant physical parameter such as the Laplace pressure, Darcy’s permeability, effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing into the pore. In determining all these parameters independently, the consistency of our analysis of such forced imbibition processes is demonstrated.
When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces, which are hard to quantify experimentally, in particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure quantifies relevant physical parameter such as the Laplace pressure, Darcy's permeability, effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing into the pore. In determining all these parameters independently, the consistency of our analysis of such forced imbibition processes is demonstrated.
Flowing fluids confined to pores with diametera on the µm to nm scale are important in many contexts: oil recovery from porous rocks [1]; separation processes in zeolithes [2]; nanofluidic devices such as liquids in nanotubes [3]; nanolithography [4], nanolubrication [5], fluid transport in living organisms [6] and many other applications [1]. However, despite its importance for so many processes in physics, chemistry, biology and technology, the flow of fluids into (and inside) nanoporous materials often is not well understood: the effect of pore surface structure on the flowing fluid [5,7,8] is difficult to assess, in terms of hydrodynamics, the problem differs dramatically from the macroscopic fluid dynamics [9,10]; and although very beautiful experiments have recently been made (e.g. [11,12]), more information is needed for a complete description of the relevant microfluidic process.
In the present work, we propose to use forced imbibition with the external pressure as a convenient control parameter to obtain a much more diverse information on the parameters controlling flow into capillaries than heretofore possible. Extending our recent study of imbibition at zero pressure [13], we concisely describe the theoretical basis for this new concept, and provide a comprehensive test of the concept in terms of a “computer experiment” on a generic model system (a fluid composed of Lennard-Jones particles flowing into a tube with a perfectly crystalline (almost rigid [13]) wall and spherical cross-section, see Fig. 1. We also provide a stringent test of our description by having estimated all parameters of the theory in independent earlier work [13]), and hence there are no adjustable parameters whatsoever. We emphasize that our procedures and analysis could be followed in experiments with real materials fully analogously.
We briefly summarize the pertinent theory. On a macroscopic scale the rise of a fluid meniscus at height H(t) over the entrance of a capillary with time t is de-
Here γ LV is the liquid-vapour surface tension of the liquid, η the shear viscosity of the fluid, R the pore radius, θ the contact angle, and H 0 a constant (which accounts for the fact that Eq.( 1) holds only after some transient time when inertial effects have already vanished). Eq.( 1) follows when one balances the viscous drag force 8η R 2 H(t) dH(t) dt with the Laplace pressure
The applicability of Eq.( 1) for ultrathin pores has been rather controversial [16,17,18]. This debate was clarified [13] by recalling that on the nanoscale the slip length δ [19,20] must not be neglected. According to the definition of this length, the drag force under slip flow conditions in a tube of radius R and slip length δ is equal to the drag force for a no-slip flow in a tube of effective radius R + δ. Thus one ends up with a modified Lucas-Washburn relationship:
On the left hand side of Eq. 2 we have now also included an external pressure term P ext . If one uses Darcy’s permeability [21] κ = (R + δ) 2 /8, Eq. 2 can be written in a form which does not depend on the capillary radius anymore, introducing also the rate v(P ext ) = dH 2 (t) dt ,
For constant P ext , Eq. 3 is easily integrated to
Eq. 3 shows that v(P ext ) varies linearly with P ext , so measuring this relationship yields both parameters P L and η/κ. Instead of using the height H(t) by observing the meniscus, one may alternatively estimate v(P ext ) from the time variation of the mass of the fluid inside the capillary (i.e. the total number of particles N (t) ∝ H(t) which has entered the capillary up to the time t). In contrast, the classical experiments on spontaneous imbibition of a fluid, where P ext = 0, yield only the product κP L , and hence even if the fluid viscosity η is known, one cannot discern the effects due to the driving force (∝ P L ) and due to the drag force (∝ κ). Moreover, Eq. 3 suggests the intriguing possibility of applying the present concepts to the most general case of porous media [1], irrespective of the particular geometry and topology of the channels in such materials, but this will not be followed up here.
We now present a test of the above concepts by a quantitative analysis of the computer experiment outlin
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