A stationary system composed of a fibre, liquid and air consists of a background for the shape determination of a typical liquid at the liquid-fibre inter-phase. Up to the present, it has not been possible to define this shape by a mathematical function. In this study a differential equation was found and solved analytically, describing the liquid curve at the mentioned interphase in the air-fiber-liquid system. This equation was solved and the result calculated by a numerical method, which were then compared with the experimental data obtained by a measurement technique developed by us

Dembický, Josef

Wiener, Jakub

Fibres and Textiles in Eastern Europe

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Dembický J., Wiener J.; Simulation of Single Fibre Wetting.FIBRES & TEXTILES in Eastern Europe  2010, Vol. 18, No.  5 (82) pp. 51-54.51Simulation of Single Fibre WettingJosef Dembický, Jakub WienerTechnical University of Liberec, Studentská 2, 461 17 Liberec, Czech RepublicE-mail:  josef.dembicky@tul.cz, jakub.wiener@tul.czAbstractA stationary system composed of a fibre, liquid and air consists of a background for the shape determination of a typical liquid at the liquid-fibre inter-phase. Up to the present, it has not been possible to define this shape by a mathematical function. In this study a differential equa-tion was found and solved analytically,  describing the liquid curve at the mentioned inter-phase in the air-fiber-liquid system. This equation was solved and the result calculated by a numerical method, which were then compared with the experimental  data obtained by a mea-surement technique developed by us.Key words: wetting, fibre, air-liquid curve, experimental verification.n IntroductionThe Wilhelmy method is one of the most important for the determination of wet-ting parameters. It is often used for the determination of liquid surface tension. The principal is based on the use of a thin plate immersed in a test liquid. On both sides of the plate a meniscus is created; the shape and maximum height of the liq-uid that rise along the plate are given by the Laplace equation. The force acting on the plate immersed in the liquid has the following value (1).F = p γLG cosθ - ρ g A d       (1)where,p  - plate perimeter, mγLG  - surface tension of the liquid, N.m-1θ  - contact wetting angle, degρ  - liquid density, kg.m-3A  - area of plate base, m-2d  - height of immersion, mIf the end of the plate is placed at the liq-uid level, then the following relationship of force F is valid(2).F = p γLG cosθ                    (2)The principal of this method is used for measuring the contact wetting angles of fibres immersed in liquid. Figure 1 presents the inter-phase of the liquid-fibre-air system, which is de-scribed by equations (3) to (11). Equation (11) presents the relationship of the ge-ometry of the liquid surface, which was developed and solved by a numerical method (see equation (12) to (16)).The Young-Dupree equation created based on analysis (3) [5]. +=21LG R1R1p γ  (3)where,p  - Laplace pressure, Pa,γLG  - Liquid surface tension, N m-1,R1, R2  - radii of the curvature, m (see Figure 2). For a liquid in a static state, the pressure at each point of the liquid is given by equation (4) [6]. p = ρ g z                     (4)where,p  - hydrostatic pressure, Pa,ρ  - liquid density, kg m-3,z  - height, m.By substituting equation (4) into eq. (3), we obtain equation (5) [5]. zkR1R121=+             (5)where,k is a constant given by equation (6).k = ρ g / γLG                    (6)For determination of the radius of curva-ture R1, equation (7) was used [1]./23/21zz1R+=                 (7)/z  - first derivative of parameter z//z  - second derivative of parameter zThe radius of curvature R2 is based on the definition of Meusnier´s theorem [1], according to which: ( )ϑcosrR 2 =                    (8)where,θ  - angle between the normal cut of the plane of a radius described by curva-ture R2 and the horizontal level, degFigure 1. Scheme of the fibre-liquid-air 3-phase system (a) and photograph of a fibre im-mersed in a liquid (b), according to [4]; Zmax - maximum height of the ascenting liquid, m, θ - contact wetting angle, deg.Figure 2. Scheme of the geometry of the liquid-solid system with radii R1 und R2; n - normal to the curvature, M - centre of the curvature radius.FIBRES & TEXTILES in Eastern Europe  2010, Vol. 18, No.  5 (82)52Table 1. Input parameters for the numerical solution of equation (11).Parameter Polyester distill waterPolyester HeptanePolyester DiiodomethaneContact angle θ, deg 69.50 29.50 42.30Fiber diameter r1, μm 14.25 15.25 14.20Maximal liquid height Zmax, μm 25.40 50.00 46.20Liquid surface tensionγLG, mN m-1 72.80 20.40 50.80Figure 3. Model (step 1 μm) and experimental data for the meniscus shape of the distilled water (a), heptane (b) and diiodomethane (c) – interaction system with PES fibre. Figure 4. Models with different steps and experimental data for the meniscus shape of the distilled water system in the contact with PES fibre. r  - horizontal distance from the fibre axis, m From a geometrical point of view, rela-tionship (9) is valid. ( ) ( )/2/z1zásincos+==ϑ  (9)where,α  - angle between the tangent and the cut of the plane of a radius described by curvature R2 and the horizontal level, degAfter the substitution of equation (9) into equation (8) we obtain relationship (10). /2/2 z1rzR1+=            (10)The differential equation sought aris-es from the substitution of equations (10) and (6) into eq. (4). Equation (11) presents the final relationship. zkz1rzz1z/2/23/2//=+++     (11)Euler’s numerical method was used to solve this differential equation. Figure 2 illustrates the geometry of the liquid-solid system with both radii of the curvature. n Description of Euler’s methodThe system is given by equation (12) [2].( )yx,fdxdy=                  (12)The initial conditions give the solution at point x0, and subsequently the solutions for points x1, . . . , xn are calculated. The value for point xi+1 will be found by the linearisation of the function at point xi. It can be stated that hi = xi+1 – xi, hence relationship (13) is then valid.( )iiiixii1i y,xfhydxdyhyyi+=+≈+  ( )iiiixii1i y,xfhydxdyhyyi+=+≈+            (13)The procedure used to solve differential equation (11) is described by the three equations mentioned below - (14), (15) and (16).[ ] +++=ii2/i2i/i//rz1zk z1z iz (14)i/i/1i/ zhzz +=+          (15)i/i/1i zhzz +=+           (16)n Results and discussionExamples of liquid meniscus simulation at the inter-phase with polyester fibre are illustrated in Figures 3.a, 3.b and 3.c. Simulated values are compared with the experimental. Euler’s method was also used for the cal-culation of parameter z. Short steps were chosen (10-6 m) in order to achieve high precision. Figure 3.c shows a compari-son of liquid curves for different steps. Particular parts of derivations 2, 1 and 0 a) b) c)yx(x,y)yx53FIBRES & TEXTILES in Eastern Europe  2010, Vol. 18, No.  5 (82)are calculated in the sequence according to equation (13).Simulation of the liquid shape in con-tact with polyester fibre Values of the input parameters are listed in Table 1. The solution of the analytically deter-mined differential equation fits very well with the experimental liquid shape. The experimental work was carried out using the measurement equipment described in [7], allowing to make a comparison of the residual variability [3] with the measured variability for experimental and simulated values (Tables 2, 3, 4). In Figure 3.c experimental and simulated values of the meniscus shape of distilled water and polyester fibre are compared.Figure 4 shows the dependence of the meniscus shape on the step. Lower steps fit better to the experimental values. After analysing the step, 10-6m was found as optimal. Figures 3.b and 3.c illustrate other exam-ples – polyester fibre in contact with hep-tane and diiodomethane. The behaviour of the model at high distances from the fibre wall is characterised in Figures 5.Table 2 presents a variability analysis of the interaction between PES-fibre and distilled water. A variability analy-sis of the other examples is presented in Tables 3 and 4 (heptane with PES-fibre and diiodomethane with PES-fibre). The residual and measured variabilities are, in all cases, very similar, hence it can be stated that good model precision was achieved. A control was obtained with the purpose of comparing the surface force calcu-lated, according to (14), with that ob-tained from the model. The integration was made at such a distance from the fibre wall that a horizontal line would Figure 5. Behaviour of the PES – liquid model: a) water 10 mm distant from the fibre wall, b)  water 7 mm distant from the fibre wall, c) diiodomethane model 7 mm distant from the fibre wall.a)Table 2. Variability of the interaction of polyester fibre – distilled water; r - distance from fibre wall in m, Ze - experimentally determined liquid height in m, Zm - liquid height cal-culated from the model.r, μm Ze, μm Zm, μm Square residuum, μm2 Variability, μm214.25 25.41 25.4 0 020.95 23.80 23.5 0.0688 0.010334.36 21.74 20.9 0.6696 0.161059.50 18.43 18.3 0.0186 0.200093.58 15.91 16.2 0.0637 0.0200123.2 14.00 14.9 0.7460 0.1830228.0 11.90 12.0 0.0148 0.1330490.0   9.30   8.6 0.5384 1.9600Sum, μm2 2.1198  -Residual variability, μm2 0.3028  -Average variability, μm2  - 0.3329Table 3. Variability of the interaction of polyester fibre – heptane. r, μm Ze, μm Zm, μm Square residuum, μm2 Variability, μm2  15.3 50.0 50.0 0 0  20.7 42.8 43.4 0.449 0.374  29.3 39.1 40.0 0.770 0.642  30.6 39.0 39.6 0.389 0.324  51.0 36.9 34.5 5.880 0.490  51.4 35.4 34.3 1.140 0.953  66.8 31.0 31.7 0.529 4.410  68.1 30.9 31.6 0.464 3.870100.2 27.4 27.9 0.236 1.970101.2 26.5 27.8 1.830 1.520132.8 26.3 25.2 1.190 0.995134.2 26.2 25.2 1.140 0.875235.0 18.1 20.0 3.550 2.730Sum, μm2 17.5695  -Residual variability, μm2   1.3515  -Average variability, μm2  - 1.4734Table 4. Variability of the interaction of polyester fibre – diiodomethane.r, μm Ze, μm Zm, μm Square residuum, μm2 Variability, μm2  14 4.62 46.2 0 0  24 3.75 39.3   3.1664 45.5  40 3.04 33.9 12.7287 7.38  70 2.71 28.5   1.8527 22.3107 2.32 24.4   7.2394 5.11220 1.70 17.7 30.5993 9.33400 1.08 12.4 21.3833 8.66Sum, μm2 76.9698  -Residual variability, μm2 12.8283  -Average variability, μm2  - 14.04b) c)FIBRES & TEXTILES in Eastern Europe  2010, Vol. 18, No.  5 (82)54Received 15.12.2008         Reviewed 16.02.2010not arise. 10 mm (Figure 5.a) and 7 mm (Figures 5.b and 5.c) were chosen as ex-amples. The differences determined can be considered as a very good match – un-der 3% (Table 5). F = 2 p r γLG cos θ               (14) where,F - wetting force, N m-1θ - contact angle, degThe simulated surface force was deter-mined by the integration of the liquid curve. The analytically calculated surface force was calculated from equation (14).n ConclusionBased on the analytical description, a dif-ferential equation, which was solved by Euler’s numerical method, was found. Before solving the initial condition, the contact angle, maximal liquid height and liquid density were determined. The model’s precision was checked and confirmed after detailed analysis of vari-ability. Comparison of the analytically calculated surface force and simulated surface force was the second check-ing procedure. The model fits well ex-perimental values, as the differences achieved in all cases are within the range of 1.1 to 2.3%. Table 5. Comparison of simulated and analytically calculated surface force with the simu-lated surface force.Liquid Simulated surface force, mNAnalytically calculated surface force, mNDifference, %Water 2.2595 2.2853 1.13Heptane 1.8343 1.8674 1.77Diiodomethane 3.2757 3.3518 2.27Knowledge of the meniscus shape has significant importance for further re-search of the wetting phenomena, hence the model established is of great benefit. References  1.  Rektorys K.: Přehled užité matematiky, SNTL 1988.  2.  Bartsch H.J.: Matematické vzorce, VEB Fachbuchverlag, Leipzig 1987.  3.  Meloun J., Militký J.: Statistické zpracov-ání experimentálních dat, PLUS , Praha 1994.  4.  Dembický J.: Messung der Netzfähigkeit von Textilfasern, Mell. Textil., 5, pp. 398-400, 2003.  5.  Adamson A.W.: Physical Chemistry of Surfaces, 5th Edition, New York 1990.  6.  Hála E., Reiser A.: Fyzikální chemie 1, Academia, NČSAV, Praha 1971.  7.  Dembický,J., Wiener,J.: Melliand Textil-berichte 86, 2005, 6, pp. 420-4228.   8.  Petrulyte S., Baltakyte R.: Fibres and Textiles in Eastern Europe, 4 (69) / 2008, pp. 62–66.4th International Conference on the Behaviour of Polymers  and Polymer-Based Nanomaterials  Related to Their Structure20-23 September 2010, Łódź, PolandOrganiser:Centre of Molecular and Macromolecular Studies, Polish Academy of Sciences, Łódź, PolandChairman of Program Committee:Professor. Andrzej Galeski, Ph.D., D.ScTopics:n Mechanical and thermal behaviour of polymers  and polymer-based nanomaterialsn Design of the microstructure of sensitive materials n Computational methods, modeling and theoryn Molecular dynamics modeling of structural transformationsn Nano-filled, crystalline and nano-structured polymer materialsn Linkage between different length scalesn Micro-mechanicsn Innovative methods for characterisation at multiscale levelsContact:Centre of Molecular and Macromolecular Studies Polish Academy of Sciencesul. Sienkiewicza, 90-363 Łódź, Polande-mail: cbmm@cbmm.lodz.pl

Simulation of Single Fibre Wetting

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Simulation of Single Fibre Wetting

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