Zhong, W
ABSTRACT
A statistical model, the Ising model, is used to simulate liquid wetting of fibrous assemblies. The adhesive energy between fiber and liquid and the cohesive energy within the liquid are calculated by applying the Lifshitz theory under the assumption that the interactions between a fiber cell and a liquid cell and between two liquid cells are dominated by Van der Waals forces. The work of extrinsic force, the surface tension, is taken into account by its contribution to the energy of the system. To test the validity of the model, wicking experiments involve a set of polypropylene filament yams with the same yam count but different fiber finenesses. Computer simulations are also made to show the relationship between the traveling height of the liquid front in the fibrous assemblies and the time taken. Experimental and simulation results show considerable accord.
Background
Wetting of fibrous assemblies is an essential issue in such applications as fabric dyeing and finishing, liquid filtration, clothing comfort, highly absorbent materials, geotextiles, etc. Accordingly, there has been extensive research on the wetting behavior of liquid in fiber assemblies.
In 1921, Washburn established a classical equation [11] describing a liquid's velocity moving up or down in a perpendicular capillary. To apply Washburn's equation to wicking studies in the 1950s, Minor [11, 12] established that the wicking height of liquid in a fiber or yam is proportional to the square root of the time, assuming that gravity is negligible as long as the wicking height is small. However, the effect of gravity cannot be neglected when the wicking height is great. Actually, the liquid column will cease to rise after certain period of time due to the balance of surface tension and gravity.
There were also many efforts to measure the parameters of those properties of liquid and fibrous assemblies that contribute to wetting and wicking behaviors. Hsieh et al. [5, 6, 7] attributed the driving force of liquid movement to the surface tension of the liquid and developed an experimental protocol to obtain the liquid-solid contact angle. Ghall [41 used Darcy's equation to describe wetting and wicking behavior. Accordingly, his work assessed and developed experimental techniques to measure capillary pressure, the driving force of liquid movement, and the permeability of fibrous assemblies, another parameter needed in Darcy's equation.
By now, most of the efforts to reveal the wetting behavior of fibrous assemblies were based on studies of macroscopic phenomena and adopted empirical methods. Only recently have researchers attempted to apply a stochastic approach to the study of wetting behavior of fibrous assemblies on the microscopic scale in order to determine the fundamentals of the phenomenon. Lukas et al. [10] used Ising's model combined with Monte Carlo simulation to study liquid-fiber interactions and the resulting wetting behavior of fiber networks. However, in order to analyze wetting behavior quantitatively by representing the properties of specific fibers and liquids in the model, there is still work to be done.
Ising first presented his model [15] in 1925 to study the ferromagnetic phase transition. If there is a one-- dimensional ferromagnet with very strong uniaxial anisotropy, the energy of the system, the Hamiltonian, may be described as
However, there is an essential difference between work done by Lukas et aL [10] and that using Ising's model. Rather than Equation 1 for the case of a ferromagnet, where the Ising variables are scheduled to be si = 1 or -1 depending on the spin pointing up or down, new sets of variables s^sub i^ = 1 or 0, F^sub i^ = 1 or 0 are adopted to study the wetting dynamics. As we know, the exchange energies associated with the magnetic moment of a pair of neighboring cells bearing same spin variable, either 1 or -1, are identical, whereas the pair with opposite spin variables takes a negative value in reference to the former. For the wetting process, however, since cohesion energy is only involved in the interaction between cells within the liquid, we do not expect to see a pair of empty cells having identical energy to the pair of liquid-filled cells if the spin variable system 1 and - 1 is maintained. This argument provides the basis for adopting the variable as 1 and 0. Reasoning in the same way, we can draw a similar conclusion for the adhesion energy term, since it is impractical to expect an empty cell with adhesion energy identical to a cell not only filled with liquid but covered with fibers, if the 1 and - 1 system is observed.
Thus, the Hamiltonian of a single cell j, referring to Figure 1, is calculated as an example in the manner shown in Table I. The total Hamiltonian for the system is the summation of all the Hamiltonians for the L X L cells.
The coefficients in Equation 6 are further discussed with a rationale that follows. Assume that Van der Waals forces dominate the interactions between fiber and liquid. According to the Lifshitz theory [3, 8, 14], the interaction energy per unit area between two surfaces can be expressed as
Experimental Verification
To test the validity of the model developed in this study, we performed a set of wicking experiments at room temperature, using test samples of three kinds of polypropylene filament yarns with the same yarn count but different finenesses of constituent filaments. Their specifications are listed in Table II. The samples were extracted with acetone in a Soxhlet extractor for 2 hours to remove surfactant, then twisted to 5 turns/10 cm to ensure a uniform cross section along the length of the sample, with the filaments in the sample evenly packed. The fact of such a weak twist maintains the validity of the assumption of a straight state of filaments within the samples. Each sample was hung vertically by a clamp and the free end was dipped into a bath containing a 1% solution of methylene blue. A small weight was attached to the free end of the sample by a hook to maintain the sample in a vertical state. The traveling height of the liquid was measured during the wicking tests. To obtain the wetting rate, the time required for the dye solution to travel upwards along the sample was recorded.
In the simulation, the plane is divided into 9 X 150 square cells. For convenience when calculating the weight (from the volume) of the liquid in a cell and the interfacial area between two cells, each cell is supposed to be a cubic. The dimension of a cell is determined according to the dimensions of the test samples. For example, in the case of sample 1, the width of a cell is equal to the diameter of the yam. Thus, each cell represents a volume of 0.282 X 0.282 X 0.282 mm^sup 3^.
Let each Monte Carlo step represent 1 second. The parameters needed in the simulation are listed in Table III. The experimental and simulation results of the wetting rate are shown in Figure 2. The constant B^sub 1^ is 3.72, and the constants k^sub 1^ and k^sub 1^ are calculated by Equation 9 as
The results of our experiments and simulations show considerable accord. Keeping the yarn count as constant, with the decreased filament diameter in the yam, the total surface area of a unit length of the yarn increases, which in turn causes an increase in the interaction area of fiber and liquid and in the interaction energy, resulting in better wettability of the yam. This outcome also agrees well with the phenomenon reported in various references.
Both the experiments and simulations show that traveling height rises substantially at first and then slows down, asymptotically approaching a plateau where the effect of the gravity of the liquid column cannot be ignored. It reveals that the Washburn equation is limited in predicting the wicking properties of fibrous assemblies.
Model Simulation
The dimensions of the cells in the lattice frame can be further defined, for example, to be equal to the diameter of a single fiber or filament in a yam, so as to increase the precision of the simulation and demonstrate the simulation of wetting of yams in more detail. The wicking process of a filament tow is shown in Figure 3. These filaments are depicted as a set of sine curves with different phases responsive to being twisted. The lattice size is 100 X 40. The width of each cell is equal to the diameter of a filament d. The parameters are B^sub 1^ = 3.72, k^sub 1^ = 1.34, k^sub 2^ = 14.2, d = 9.2 (mu)m, and beta = 1/4.
We see that the traveling rates of liquid vary in different areas. This agrees with the fact that the liquid front in a tow is not always a straight line in experiments. Liquid tends to rise more quickly in areas where packing densities are higher. In areas where packing densities are quite low, empty holes tend to form, indicating an imperfect wetting state. The simulation also shows that the traveling rate of liquid slows down with time, and the width of the liquid column decreases with height due to the balance of surface tension and gravity.
This example indicates that the method described in the study is useful when used to predict the wicking behavior of various yarns. The model can easily be modified to adapt to investigations of more complicated cases, such as liquid wetting of more complex patterns of structures, various kinds of fabrics or nonwovens, or the flow behavior of fluid through porous media.
The two-dimensional Ising model may be further modified into a three-dimensional model, in which the three-dimensional structures of yams or fabrics and their interactions with liquid in spaces can be represented to the full, and the differences that still exist between the experiment and simulation will be depressed further.
Conclusions
From the point of view of statistical thermodynamics, wetting behavior is a reflection of the interactions and the resulting balance between fiber and liquid cells that constitute the system. We use a statistical model, the two-- dimensional Ising model, combined with Monte Carlo simulation, to describe the wetting process, which is regarded as the spaces between fibers changing from a gas-dominant state to a liquid-dominant state. Such changes are driven by the difference in energy between the two states and finally terminated by the balance of surface tension and gravity.
To accommodate Ising's model to the process of liquid wetting in fibrous assemblies, we have adopted new sets of variables, s^sub i^ = 1 (liquid cell) or 0 (gas cell), F^sub i^ 1 (fiber cell) or 0 (cell without fiber), to study the wetting dynamics. Characteristics of interacting fiber materials and liquid can be represented by the coefficients of the adhesive energy between fiber and liquid and cohesive energy within the liquid. We account for the work of extrinsic force-surface tension-by its contribution to the energy of the system derived from liquid and the fibrous assemblies. The results of our simulation are in good agreement with the wicking experiments, indicating good prospects for the method to be used in this area. The dimension of a cell in the lattice frame is further defined to be equal to the diameter of a single filament in a tow, so as to increase the precision of the simulation as well as to better demonstrate the simulation of wetting yarns.
The model can be modified to adapt to liquid wetting of more complex patterns of structures by altering the initial configuration of fiber assemblies and the input parameters of different fiber types and liquids. The simulation results can be further improved by expanding the model three-dimensionally.
Literature Cited
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Manuscript received October 11, 2000; accepted December 12, 2000.
W. ZHONG,1 X. DING,1,2 AND Z. L. TANG3
China Textile University, Shanghai 200051, People's Republic of China
1 College of Textiles.
2 To whom the correspondence should be addressed.
3 College of Material Science and Engineering.
