Diffusion of dyes in polyester fibers. III. Time delay in establishment of the equilibrium concentration of dye at fiber surface

In the dyeing process of the copolyester fiber Dilana with the disperse dye Synthene Scarlet P3GL, a deviation from the simple model of Fickian sorption occurs. It manifests itself as a time delay in establishment of the equilibrium dye concentration at the fiber surface. As has been stated, the variation of C_s (the surface dye concentration) with time of dyeing fits the equation; C_s = C_∞ (1? e^?βt). Regarding the relation of C_s to t, the diffusion coefficient of the dye in the studied fiber has been calculated by the theoretical equation reported in Crank's monograph. It has been proved that the experimental data on kinetics of dyeing the fiber Dilana with Synthene Scarlet P3GL fit considerably better the tested equation than the classical Hill's equation.     

The influence of polymer and dye characteristics on diffusion in nylon 66

A study has been carried out of the relationship between polymer orientation and dyeing behavior using a specially prepared range of nylon 66 samples varying only in orientation and five dyes normally showing varying degrees of sensitivity to physical variations in nylon. Integral diffusion coefficients have been calculated using Crank's finite dyebath equation from appropriate rate of dyeing data. Evidence has been produced which suggests that except with dyes of low molecular weight, i.e., dyes insensitive to physical variations in the nylon, a diffusional interaction between diffusing dye molecules exists which leads to a variation in the concentration dependence of the diffusion coefficient with fiber orientation. The length of the diffusing dye ion has been shown to be of decisive importance in this interaction.     

Diffusion of dyes in polyester fibers. II. Diffusion coefficients from the radial distribution curves

The computational method of a concentration-dependent diffusion coefficient D(C) of dyes in fibers has been presented. This method is based on concentration profiles determined with the microphotometric technique and the numerical solution of Fick's second law of diffusion for the cylindrical system. Exploiting the grid method and data of experimental concentration profiles, diffusion coefficients of disperse dye Synthene Scarlet P3GL in the anionically modified polyester fiber have been calculated. The results have been compared with those obtained by the Boltzmann–Matano method. It was stated that (1) in the investigated polymer–dye system the relation between D and C is of the form D(C) = D₀ exp(δC); (2) the allowance for the cylindrical symmetry of the fiber leads to the lower values D(C) in the total concentration range than those obtained by the Boltzmann–Matano method; and (3) values of D₀ calculated with both methods are coincident.     

Diffusion of dyes in polyester fibers. IV. Anomalous sorption of disperse dye synthene scarlet P3GL

The investigations on dyeing of poly(ethylene terephthalate) fibers with disperse dye Synthene Scarlet P3GL have revealed anomalous sorption of the dye in heat-treated samples. In the mathematical solution of the process it has been assumed that anomalous sorption can be treated as superposition in time of two stages of Fickian sorption. The contents of absorbed dye at quasiequilibrium C_I, at the final equilibrium C_II, and the apparent diffusion coefficients D_I, and D_II have been calculated using Hill's equation.     

Diffusion of dyes in adsorptive dyeing

The concentration dependence derived from both steady-state permeation and unsteady-state dyeing runs for a system of acid dye C. I. Acid Blue 182-nylon 6 film was analyzed in terms of parallel diffusion with simultaneous multimodal adsorption proposed in the previous paper. The present dyeing process was governed by the surface diffusion incorporating three kinds of Langmuirean adsorption modes. The concentration dependence of the diffusion coefficient of the cationic dye C. I. Basic Red 22 in the anionically modified polyester fiber Dilana measured by Ostrowska et al. [B. Ostrowska, A. Narebska, and H. Krzystek, J. Appl. Polym. Sci., 26 , 463 (1981)] was satisfactorily interpreted by the parallel diffusion incorporating two kinds of Langmuirean adsorption modes.     

Concentration-dependent diffusion of dye in adsorptive dyeing systems

A concentration-dependent dye diffusion model is extended in the present study to represent adsorptive dyeing systems. It is a revision of the previous one by incorporating the effect of external mass transfer resistance and concentration dependence of diffusion coefficient on the dye's diffusion process. The theoretical predictions using the extended model show excellent agreement with the observed data, significantly better than that of the previous one. It is also found that an adsorption equilibrium is achieved under most operating conditions due primarily to the fact that the dye diffusion within the pores is a considerably slower process than the dye adsorption.     

The use of reflectance measurements in the determination of diffusion of reactive dyes into cellulosic fiber

Reactive dye fixation and color yield of a dyed cellulosic fiber significantly depend on the dye diffusion into the fiber polymer system. In case of pad‐dyeing processes, dye diffusion exerts a more significant influence on dye fixation and hence color yield. This article proposes a new method for determining the extent of diffusion of reactive dyes into the fiber in pad dyeings using Kubelka–Munk equation. The K/S values are used as in an equation, %D (extent of dye diffusion) = 100 ? [(K/S_{diffusion index})/ (K/S_reference) × 100]. The article introduces and explains how to determine the new K/S variables used in this equation. The new method is simple, nondestructive, relatively faster, and applicable to industrial dyehouses, and was validated by a microscopic analysis of dyed fiber cross‐section carried out in this work and to the dye manufacturer's recommendations for dyebath‐ingredient concentrations. © 2012 Wiley Periodicals, Inc. Col Res Appl, 39, 63–69, 2014     

Diffusion/adsorption model of cellulose dyeing. II. Ordinary cellulose–direct dye system

The diffusion and adsorption of C.I. Direct Yellow 12 and Blue 15 in water-swollen ordinary cellophane sheets were examined at various ionic strengths. The concentration dependence of apparent diffusion coefficients, D_c, for these dyes was obtained from the diffusion profiles in the substrate, which were measured by the use of the cylindrical film roll method. The decrease of apparent porosity with an increase in the amounts of adsorption was observed. To explain the diffusion/adsorption behaviors of these systems, a variable porosity model was proposed and was applied to analyze the concentration dependence of D_c's. The diffusion/adsorption behaviors of these dyes could be quantitatively described by this model at relatively low ionic strengths. At higher ionic strengths and/or lower values of C, i.e., at the large values of C_im/C_m, where the C's are the concentrations of immobilized (suffix im) and mobile (suffix m) species, it needed to introduce the concept of dynamic equilibria which occurred simultaneously with diffusion but deviated from the true equiliblia measured by the adsorption experiments.     

Diffusion/adsorption model of cellulose dyeing. I. The diffusion through never-dry cellulose

The diffusion coefficients D₀, of C.I. Direct Yellow 12, Red 2, Blue 1, and Blue 15 in aqueous NaCl solution were measured at 90°C by the diaphagm cell method. The values of D₀ for Yellow 12 and Red 2 showed a salt concentration dependence and those for Blue 1 and Blue 15 were constant over the ionic strength range from 0.01 to 0.10. In the adsorption/diffusion models proposed so far for the direct dye–cellulose system, the Standing–Warwicker–Willis model was shown to be similar in principle to the Weisz–Zollinger model. The adsorption/diffusion behaviors in never-dry cellophane sheet for C.I. Direct Yellow 12 and Blue 15 were examined by the method of cylindrical film roll at 90°C. The concentration dependence of the apparent diffusion coefficient for these dyes showed an incomplete validity of both the models.     

Spectrophotometric investigation of laser dye diffusion in styrene–acrylonitrile copolymer solutions

An ultraviolet spectrometer was used to measure the absorption kinetics of a laser dye, 1,4‐bis(5‐phenyloxazoyl‐2‐yl) benzene, at its maximum wavelength in solutions of styrene–acrylonitrile copolymer dissolved in 1,4‐dioxane. From the time‐dependent absorbance data, the dye diffusion coefficient was calculated by a combination of Fick's law of diffusion with the Beer–Lambert law of absorption. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 95: 1481–1484, 2005     

Diffusion with simultaneous reaction of reactive dyes in cellulose

The theoretical equations which described the concentration profile of immobilized species of reactive dyes in the substrate were derived from the diffusion equation with the chemical reaction of first-order or pseudofirst-order. The theoretical profiles in the substrate described by the equations were discussed. The larger the diffusion coefficient D of active species and the smaller the reaction rate constant k, the deeper is the penetration of the immobilized and active species of reactive dyes into the substrate. The method of estimating D and k from the diffusion profiles of both species obtained by means of the method of the cylindrical cellophane film roll was described. The diffusion coefficients of the hydrolyzed species of C.I. Reactive Orange 1 and Red 1 were nearly constant in all the pH values examined. The concentration profiles of both the species of Orange 1 at pH 8.8 were identical with the theoretical ones; while the profiles of immobilized species of Red 1 at pH 10 and of Orange 1 at pH 12 agreed with the theoretical ones and those of active species did not because of the hydrolysis. The diffusion coefficients of active species of these dyes at these pH ranges were smaller than those of the hydrolyzed species.     

共聚酯共混改性丙纶染色动力学研究

以阳离子蓝X-GRRL为染料,根据100℃时,常压阳离子染料可染共聚酯共混改性丙纶在不同时间的上染量,采用无限染浴模型计算染料分子在改性丙纶中的扩散系数,据此求出染色过程的扩散活化能。其结果如下:100℃时阳离子蓝X-GRRL染料在改性丙纶中的扩散系数为1．8×10 -11cm2／s,是丙纶的扩散系数的10 000倍以上;扩散活化能为89．13 kJ／mol。     

Reaction and diffusion of reactive disperse dye in nylon 6

The diffusion of a reactive disperse dye with a vinylsulfonyl group accompanied by simultaneous reaction with the amino end groups in nylon 6 was examined by the method of cylindrical film roll at 70°C and pH 2.2–8.0. The experimental diffusion profiles of the active and fixed species of the dye in nylon 6 were confirmed to be described by the diffusion equation accompanied by the chemical reaction with substrate taking the limited amount of the end groups into account, where the active species of dye were assumed to react only with the free base of amino end groups. The completion of the reaction with the amino end groups was observed in the first layer from the surface at pH 6.0–8.0. The value of diffusion coefficient was constant (8.0 × 10^?10 cm²/s) at all the pH's. The product of the second-order rate constant, k₂, of reaction of the dye and the dissociation constant, K_a, of the amino end groups was constant (k₂K_a = 4.0 × 10^?9 s^?1) at pH 2.2–8.0. The k₂ values of the reaction with various substrates for vinylsulfonyl and monochlorotriazinyl-reactive dyes were compared and the practical dyeing conditions were discussed.     

Estimation of Diffusion Coefficients Based on Adsorption Measurements in Model Extraction Systems

The drop volume technique has been used to measure the equilibrium and dynamic interfacial tension at the liquid/liquid interfaces of selected hydroxyoximes, as examples of chelating type hydrophobic metal ion extractants. The measurements for the kinetics of adsorption enable the calculation of diffusion coefficients. In this paper, new methods for the estimation of the diffusion coefficient by Fick's and the Maxwell‐Stefan equations are presented. The calculated values of the diffusion coefficient were compared with the values obtained from the short and long time approximation models of the Ward‐Tordai equation. The influence of the organic phase and the addition of non‐organic electrolyte to the aqueous phase on the estimated values of the diffusion coefficients were investigated.     

Polynomial Approximations Describing Rate of Dyeing from a Finite Bath

Three polynomial approximations to Wilson's equation were derived by means of Hastings's method. One of them is an approximation to Crank's equation with 3–figure accuracy and it approximates to Wilson's epuation very closely for high values of fractional equilibrium exhaustion (E), E > 0. 95. Two other polynomials approximate to Wilson's equation very well in the ranges 0. 95 > EX). 76 and 0. 76 > E > 0. 30 respectively. The polynomials derived here only include up to second order terms, so that by using the equations which are obtained by solving the polynomials for diffusion coefficient (D), D can easily be calculated from experimental dye uptake data. By using the polynomial approximation, the changes in rate–of–dyeing of a non–ionic dye on a nylon 6 fibre by addition of a dispersing agent were calculated. Results of the experiment agreed well with the calculated.     

Diffusion of Fibers in a Tubular Flow

Based on an ellipsoidal particle model, the equivalent diameter for the slip correction, diffusion coefficient, and diffusion diameter of fibers were obtained from the adjusted sphere method of Dahneke. The diffusion coefficient calculated for polydisperse crocidolite fibers compared favorably with available experimental data. Deposition of fibers in a tubular flow was then calculated with the use of the derived diffusion coefficient and applied to the human lung airways. The effect of velocity shear on particle orientation was also considered. It was found that the velocity shear had only a small effect on deposition. For a given fiber size, deposition increased in the lung distally, but at the same fiber diameter, the efficiency decreased with increasing aspect ratio.     

Studies in the aggregation of disperse dyes

The aggregation of disperse dyes in aqueous solution along with surfactants has been studied spectrophotometrically by Klotz's method and polarographically employing llkovic's equation by diffusion coefficient method. The extent of complex formation between dye and surfactants have been studied and compared with the determination of aggregation numbers from polarographic measurements.     

Concentration Dependence of the Diffusion Coefficients of Disperse Dyes in Polyethylene Terephthalate

Diffusion coefficients of disperse dyes have been calculated by Matano's method from diffusion profiles in polyethylene terephthalate(PET). The diffusion coefficients (D_c) clearly indicate concentration-dependence. It has been found, from the densities of the dyed PET and the dye, that there is no overall change in the volume of the PET and dye in dyeing. The instrinsic diffusion coefficients have been calculated from D_c and the volume fraction (φ₁) of penetrant in the amorphous region. From these values, thermodynamic diffusion coefficients (χ?) were obtained. It has been found that the relation between 1/log (χ?/χo) and 1/φ₁ is a straight line, and it therefore seems reasonable to assume that the diffusion of disperse dyes in PET can be described by Fujita's ‘free volume theory’ for the diffusion of low molecular-weight organic compounds in an amorphous high-polymer.     

Diffusion of disperse dye in atactic polystyrene

Sublimative desorption experiments were carried out on atactic polystyrene containing p-nitroaniline, p-aminoazobenzene, or C.I. Disperse Yellow 7 at 114°–170°C (above the T_g of the polymer). The diffusion coefficient of each dye in the polymer increased monotonically with rise in the desorption temperature. The mode of this change was exactly expressed by a WLF relation having the universal parameters given for amorphous polymers. The value of B_d, defined as the ratio of diffusional penetrant volume to that of a segment of the chain molecule, varied from 0.37 to 0.70 for the different dyes used. It is also shown that the B_d value is related to the logarithmic rotational volume of the dye molecule estimated from a molecular model.     

Temperature dependence of diffusion of a disperse dye in the multiple layers of nylon fabric

The apparent diffusion coefficients of a disperse dye in the multiple layers of nylon fabric at 45-75°C are determined as the dyeing model of the textile assembly. The modified pore model is introduced to explain the diffusion behavior in the multiple fabric. The Arrhenius plot of the diffusion coefficient of dye in the spaces or pores of nylon fabric gave a straight line and the activation energy of diffusion was about 9.4 kcal/mol. The temperature dependence of the dye diffusivity in polymer matrix was quantitatively described by a WLF-type relation, where a_T is a shift factor of diffusion and T_s is the reference temperature.