The Kinetics of Wicking of Liquid Droplets into Yarns

The kinetics of wicking of liquid droplets into yarns was studied by a computerized imaging system. A new method is suggested for characterization of the yarn structure by monitoring the droplet absorption. The method is based on the comparative analysis of the time needed for the droplet disappearance as a function of the droplet volume for various yarns. A mathematical model is developed for the description of the wicking kinetics. We show that for wetting liquids the time of droplet absorption Tw is a linear function of the initial droplet volume squared V0 . For a given liquid-yarn pair, the slope of this relationship provides important information about the yarn properties. The linear relationship between Tw and V0 2 has been verified by the experimental data with a typical spin finish. The model predicts that droplet wicking could occur even if the advancing contact angle θa is slightly greater than 90°. However, for non-wetting liquids the relationship between Tw and V0 2 is non-linear. A criterion for droplet wicking into non-wetted yarns is obtained. 2 The Kinetics of Wicking of Liquid Droplets into Yarns XUEMIN CHEN, KONSTANTIN G. KORNEV, YASH K. KAMATH, AND ALEXANDER V. NEIMARK TRI/Princeton, P.O. Box 625, Princeton, New Jersey 08542, U.S.A. Textile fibers and yarns are often treated with spin finishes that act as lubricants and antistatic agents during processing. Deposited on the yarn surface, the finish liquid wicks into the interfiber space thus providing the filament cohesion and modifying the yarn mechanical and chemical properties [12]. To process the yarns satisfactorily, absorption of the finish liquid by yarns must be sufficiently fast and uniform. To evaluate the performance of spin finishes or to characterize the wettability and structural properties of yarns, standard techniques of wicking is used [2, 4-8, 10, 13]. In these techniques, the yarn is either partially immersed in a liquid reservoir [3,5,7] or a constant supply of liquid is delivered to a certain point on the yarn [4]. The position of the liquid front is traced as a function of time. The relation between the liquid front position and the time of droplet wicking is assumed to be of the Lucas-Washburn type [6,13], t 2 cos R L a 2 η θ γ = (1) where L is the liquid front position or wicking length, γ and η are the surface tension and viscosity of the liquid, respectively, θa is the apparent contact angle, R is the effective hydraulic radius of the interfilament pores, and t is time. In this paper, we present a new method to study the wicking in yarns by recording the time needed for the complete absorption of a droplet deposited on the yarn surface. We have developed a PC-based imaging system and a mathematical model to study the wicking of liquid 3 droplets into yarns. The selected images given in Figure 1 show the dynamics of a hexadecane droplet absorption in a polypropylene (PP) yarn. For a given liquid-yarn pair, the time of droplet disappearance depends only on the initial droplet volume. The mathematical model allows us to relate the time of droplet absorption to droplet volume. Unifying the wetting and non-wetting cases, the model operates with two driving forces responsible for wicking. The pressure drop at the liquid front causes the imbibition of a wetting liquid and hinders the penetration of a nonwetting liquid; The Laplacian pressure difference due to droplet curvature facilitates the liquid penetration for both wetting and non-wetting cases. Thus, the model generalizes the Washburn approach and enables us to obtain a quantitative criterion for droplet wicking. The model is tested with experimental data to show its applicability to wicking of liquids in yarns. Theory THE MODEL When deposited on a yarn, a droplet of wetting liquids spontaneously wicks into the yarn due to the capillary forces associated with the given structure and geometry of the void spaces between the filaments. The model describing the droplet disappearance focuses on the stage just after the yarn section underneath the droplet has been saturated with the liquid (Figure 2). When the droplet size is sufficiently small, the effect of gravity on the droplet shape is negligible and the droplet may be considered as an axisymmetric one. Quantitatively, the approximation is valid if the characteristic length of capillary wave, g cap ρ γ / = l is larger than the droplet radius d R . Typically, parameter cap l is of the order of millimeter, therefore the smaller droplets are the objects of our study. 4 The movement of the liquid front along the yarn is caused by a pressure difference between the droplet, Pd, and the liquid front, Pf. Since the process of liquid imbibition is slow, we can assume that the droplet takes an equilibrium shape at each instant of time. Then the pressures can be expressed via the droplet curvature, 2H (see the Appendix for its definition), and the hydraulic radius of the yarn pores, R, as Pd = Pg + (2H)γ (2) Pf = Pg – (2γ cos θa) /R (3) where Pg is the atmospheric pressure. Applying Darcy’s law [5, 7,11] for the flow rate, we have , L R cos H k 2 L P P k dt dL a f d η θ γ η ) ( ) ( + = − = (4) where k is the permeability of the yarn. Equation (4) must be complemented by i) a condition of mass conservation and ii) an expression for the droplet curvature. i) The condition of mass conservation Assuming that the evaporation of the liquid is negligible and the liquid is incompressible, the mass balance can be written as Vd + επRy(Ld + 2L) = V0 + επRyL0 = Vtotal = constant (5) here Vd is the current droplet volume, Ld is the current droplet length (see Figure 2 and the Appendix for their definitions) V0 and L0 are the droplet volume and length at the initial instant 5 of time t = 0, Ry is the yarn radius, Vtotal is the total liquid volume, and ε is the yarn porosity. Introducing the notations ∆Vd = Vd V0 , (6) ∆Ld = Ld L0 , (7) the volume of liquid that has been absorbed by the yarn, Vw, can be expressed as, ( ) L L R V V d y d w 2 2 + ∆ = ∆ − = επ . (8) ii) The droplet curvature as a function of droplet volume The equilibrium shape of a droplet can be specified by four parameters: the yarn radius, Ry, the maximum radius of the droplet, Rd, the droplet length, Ld, and the contact angle at which the droplet meets the yarn, θ [1], see Figure 2. The formulas for droplet volume and droplet curvature can be found in the Appendix. Because of the dynamic nature of the wicking process and the complex geometry of the yarn surface, θ and θa may differ. GENERAL SOLUTION Making use of equations (5)-(8) to express the moving front coordinate through the droplet volume and the droplet length as 2 2 2 y d y d R L R V L επ επ ∆ + ∆ − = , (9) and introducing the dimensionless droplet volume, d V , and droplet length, d L , as