THE THEORY OF DYEING

Assignment By Professor : Yasir Ansari

Name : Ahmer Adnan Baloch

Colour PerceptionWe see colour with the sensors in the retina of the eye called rods and cones. The rods are sensitive to low light and the cones, which require a greater intensity of light, are sensitive to colour. The message is passed to the optic nerve and then on to the brain. We see colour because of the Rods and Cones in our eyes: There are about 12million rods and about 6 to 7 million cones, in the human eye. Rods are more sensitive than the cones but they are not sensitive to colour, they perceive images as black, white and different shades of grey. More than one thousand times as sensitive, the rods respond better to blue but very little to red light. Each cone contains one of three pigments sensitive to either RED GREEN or BLUE. Each pigment absorbs a particular wavelength of colour. There are short wavelength cones that absorb blue light, middle wavelength cones that absorb green light, and long wavelength cones that absorb red light. When we observe a colour that has a wavelength between that of the primary colours red, green and blue, combinations of the cones are stimulated. An example could be that yellow light stimulates cones that are sensitive to red and to green light. The result is that we can detect light of all colours in the visible spectrum. People who suffer colour blindness have less numbers of particular cones than normal, so they get colours confused. If we lose our eye sight, the body adapts and receives colour rays through the skin. It takes time for the body to adapt, but it has been shown that people who are blind, can differentiate between different colours. The eye picks up colour and light by the Rods and Cones. It is the Cones that detect Colour. Each cone contains one of three pigments sensitive to either RED GREEN or BLUE. Dye A dye can generally be described as a colored substance that has an affinity to the substrate to which it is being applied. The dye is generally applied in an aqueous solution, and may require a mordant to improve the fastness of the dye on the fiber.

AuxochromeThis is a group of atoms attached to a chromophorewhich modifies the ability of that chromophore to absorb light. Example-OH , - NH2, Aldehydes. An auxochrome is a functional group of atoms with nonbonded electrons which, when attached to a chromophore, alters both the wavelength and intensity of absorption. If these groups are in direct conjugation with the pi-system of the chromophore, they may increase the wavelength at which the light is absorbed and as a result intensify the absorption. A feature of these auxochromes is the presence of at least one lone pair of electrons which can be viewed as extending the conjugated system by resonance. It increases the color of any organic compound. For example, benzene does not display color as it does not have any chromophore but nitrobenzene is pale yellow color because of the presence of nitro group. Para-hydroxynitrobenzene exhibits a deep yellow color. Here an auxochrome (-OH) is conjugated with the chromophore -NO2. Similar behavior is happens in azo benzene (red color) but para-hydroxy azobenzene is dark red color. There are mainly two types of auxochromes1. Acidic -COOH, -OH, -SO3H 2. Basic -NHR, -NR2, -NH2 The presence of an auxochrome in the chromogen molecule is essential to make a dye. However, if an auxochrome is present in the meta position to the chromophore, it does not affect the colour. An auxochrome is known as a compound that produces red shift as it increases the wavelength of absorption therefore moving closer to infrared light.

ChromophoreA chromophore is part (or moiety) of a molecule responsible for its color. When a molecule absorbs certain wavelengths of visible light and transmits or reflects others, the molecule has a color. A chromophore is a region in a molecule where the energy difference between two different molecular orbitals falls within the range of the visible spectrum. Visible light that hits the chromophore can thus be absorbed by exciting an electron from its ground state into an excited state. In biological molecules that serve to capture or detect light energy, the chromophore is the moiety that causes a conformational change of the molecule when hit by light. Chromophores almost always arise in one of two formsconjugated pi systems or resonating systems. and metal complexes.[contradiction] In the former, the energy levels that the electrons jump between are extended pi orbitals created by a series of alternating single and double bonds, often in aromatic systems. Common examples include retinal (used in the eye to detect light), various food colorings, fabric dyes (azo compounds), lycopene, -carotene, and anthocyanins. The metal complex chromophores arise from the splitting of d-orbitals by binding of a transition metal to ligands. Examples of such chromophores can be seen in chlorophyll (used by plants for photosynthesis), hemoglobin, hemocyanin, and colorful minerals such as malachite and amethyst. A common motif in biochemistry is chromophores consisting of four pyrrole rings. These come in two types: * the pyrroles form an open chain, no metalphytochrome, phycobilin, bilirubin * the pyrroles form a ring (porphyrin), with a metal in the centerheme, chlorophyll

Color abstractionsThe foundations of pre-20th-century color theory were built around pure or ideal colors, characterized by sensory experiences rather than attributes of the physical world. This has led to a number of inaccuracies in traditional color theory principles that are not always remedied in modern formulations.[citation needed] The most important problem has been a confusion between the behavior of light mixtures, called additive color, and the behavior of paint or ink or dye or pigment mixtures, called subtractive color. This problem arises because the absorption of light by material substances follows different rules from the perception of light by the eye.

A second problem has been the failure to describe the very important effects of strong luminance (lightness) contrasts in the appearance of surface colors (such as paints or inks) as opposed to light colors; colorssuch as grays, browns or ochres cannot appear in light mixtures. Thus, a strong lightness contrast between a mid valued yellow paint and a surrounding bright white makes the yellow appear to be green or brown, while a strong brightness contrast between a rainbow and the surrounding sky makes the yellow in a rainbow appear to be a fainter yellow or white. A third problem has been the tendency to describe color effects holistically or categorically, for example as a contrast between yellowand blueconceived as generic colors, when most color effects are due to contrasts on three relative attributes that define all colors: 1. lightness (light vs. dark, or white vs. black), 2. saturation (intense vs. dull), and 3. hue (e.g., red, orange, yellow, green, blue or purple). Thus, the visual impact of yellowvs. bluehues in visual design depends on the relative lightness and intensity of the hues. These confusions are partly historical, and arose in scientific uncertainty about color perception that was not resolved until the late 19th century, when the artistic notions were already entrenched. However they also arise from the attempt to describe the highly contextual and flexible behavior of color perception in terms of abstract color sensations that can be generated equivalently by any visual media. Many historical color theorists have assumed that three pure primary colors can mix all possible colors, and that any failure of specific paints or inks to match this ideal performance is due to the impurity or imperfection of the colorants. In reality, only imaginary primary colors used in colorimetry can mixor quantify all visible (perceptually possible) colors; but to do this the colors are defined as lying outside the range of visible colorsthey cannot be seen. Any three real primary colors of light, paint or ink can mix only a limited range of colors, called a gamut, which is always smaller (contains fewer colors) than the full range of colors humans can perceive.

Dyes In SolutionAt some stage in all dyeing processes the color chemicals for sorption by the goods must be in the finest state of subdivision possible, which is monomolecular, i.e., single molecules or molecular ions which can be negatively or positively charged (anions or cations respectively). The molecular ions are almost always dissolved in water, but for an exception, refer to the upcoming sections on disperse dyes and their application by continuous dyeing methods: By contrast, dispersions of even the most finely ground pigment contain particles which are gross, compared with the size of dye molecules or ions, and they do not diffuse into fibers. Dyes are exposed to fibers by the circulation of dye bath and fiber relative to one another. Given the right conditions, the molecules or ions can make their way through the water to the fiber surfaces, down pores, cracks and fissures in the fiber surfaces, and ultimately between the polymer molecules, becoming distributed in the fiber itself. But these dye solutions are not as simple as they might seem.

AggregationIn the case of dyes for cellulose, the finest state of subdivision has the color bearing chemicals in the form of anions, written Dye', each with its accompanying number (n) of sodium cations, n Na*, foroverall electrical neutrality; i.e., equal numbers of positive and negative charges. Here n represents the number of negative charges perdye anion and is usually between two and four. When dye solutions are dilute and the concentrations of salts (electrolytes) are low these anions are the principle form of the dye. However; as the con-centrations of dissolved dye and electrolyte are increased, the an-ions become attracted to one another by weak forces, known as vander Waals forces, forming aggregates or larger clusters. ln extreme cases these may grow large enough to become insoluble and then the aggregate is actually precipitated out of the water. Dye manufacturers rely on this phenomenon to salt out water soluble soliddye from its concentrated solutions by the addition of salt.Although it is the single dye-anion which does the dyeing, this anion is often accompanied in solution by a reservoir of soluble dye-ion to dye-ion aggregates. These are in a state of dynamic equilibrium with the single dye-anion and will break down again if dye bath conditions become right. This situation can be written:

The arrows indicate a process capable of going both ways, and sub-script m on the left hand side of the equation is the number of dyeions in the aggregate, the value of which is unknown in most cases.Aggregation is reduced as the

temperature increases, while thesolubility of most dyes increases.

Chemical PotentialThe concept of chemical potential is introduced to help explainthe transfer of direct dyes or other anionic chemicals from thedyebath to the fiber, particularly in the presence of an excess of oppositely charged ions such as the sodium cation, Na*, which is introduced by the addition of simple sodium salts. Chemical potential is analogous to electrical potential or temperature. In all three cases, when there are two masses at different po-tentials (or temperatures) and they are brought into contact, someof the potential will be transferred from the mass at the higher potential to that at the lower potential. If the potential of dye dissolvedin a dyebath in contact with the fiber is higher than that of the dyein the fiber; then dyeing or sorption will take place. If the reverse istrue, stripping or desorption of dye will take place. When dye in the dyebath and dye in the fiber are at the same chemical potential,there will be no net transfer of dye, even though the system will bein dynamic equilibrium, with dye ions constantly entering and leaving the fiber at the same rate. For a given dye solution in water let the concentration of dye-anions and sodium cations be written [Dye]s and [Na*]s, where thesquare brackets denote concentration and subscript s indicates ionsin solution. Physical chemists express these concentrations in terms of gram molecules or gram ions per literbut that does not changethe argument, only specifies the units in a more scientific wayIn hot dilute dyebaths, where aggregation is of least concem, the chemical potential of the solution is related to the product:

As tl1is product increases, so does the chemical potential of the solution. Notice that the concentration of the sodium ion is raised tothe power n making it relatively more important the higher the value of n. Increasing the sodium ion concentration by adding sodium salts to this dye solution will also increase the product and therefore the chemical potential of the solution.In an anionic dyeing situation, as with all cellulose dyeing, rais-ing the chemical potential of the dyebath by adding salt will havethe effect of driving the dye from the higher potential and into thefiber where the chemical potential will also increase until it reaches.

Solubility ProductIn saturated dye solutions-i.e., in pure dye solutions in which nomore dye will dissolve the product is called the solubility product:

Adding more sodium ions by add-ing salt to such solutions would cause the product to increase andto exceed the solubility product. To overcome this tendency somesolid dye will be precipitated out (salted out) of solution thereby reducing the concentration of dye-ion in solution and readjusting the product. If the salt concentration is high enough, only a small amount of dye anion can be present in the solution without it being salted out.It should now be apparent that the addition of simple sodiumsalts to solutions of the sodium salts of color bearing anions willlower the attraction of the anion for the waten and the anion will tryto leave the water for any available, alternative location-for ex-ample the fiber.

The Fiber-Water BoundaryAll fibers when immersed in water acquire a negative electrical po-tential in the immediate vicinity of the surface. This potential is con-sidered to be effective over a longer range than atomic dimensions.The origin and nature of the negative surface potential of fibers inwater; its relationship to streaming potential, zeta potential, the Stemlayer; the diffuse layer, the Dorman and Gouy-Chapman models ofthe diffuse layers are all beyond the scope of this work, but can eas-ily be found.It is mainly important to recognize that such a nega-tive potential will offer resistance to the approach of similarlycharged ions such as the dye anion, just as the north pole of a magnet will resist the approach of the north pole of another magnet. charged ions such as the dye anion, just as the north pole of a mag-net will resist the approach of the north pole of another magnet.Negative potential at fiber surfaces will be attractive to oppositely charged ions or cations, which is why so many fiber surfaces can be stained by basic (cationic) dyes.The negative potential of the fiber is masked (or swamped) bythe presence in the dyebath of high concentrations of such electro-lytes as salt, with its many sodium and chloride ions; and the elec-trolyte simultaneously raises the chemical potential of the dye insolution. As a result, the effectiveness of the potential barrier to re-sist dye moving to the fiber surface is doubly diminished by thepresence of simple, soluble salts. The situation is further improvedby stirring the dyebath to provide a good flow of solution across thefiber surfaces. This can be seen as eroding the boundary layer inwhich the negative potential is effective.

The Physical Dye-Fiber BondOne of the most poorly understood phenomena in dyeing is thehigh substantivity of direct clye anions and the anions of vat andsulfur dye leuco compounds for cellulose, and the causes of the physical dye-fiber bond. The anions from dissolved reactive dyesand azoic coupling components have less substantivity charged ions such as the dye anion, just as the north pole of a mag-net will resist the approach of the north pole of another magnet.

Negative potential at fiber surfaces will be attractive to oppositelycharged ions or cations, which is why so many fiber surfaces can bestained by basic (cationic) dyes.The negative potential of the fiber is masked (or swamped) bythe presence in the dyebath of high concentrations of such electro-lytes as salt, with its many sodium and chloride ions; and the elec-trolyte simultaneously raises the chemical potential of the dye insolution. As a result, the effectiveness of the potential barrier to re-sist dye moving to the fiber surface is doubly diminished by thepresence of simple, soluble salts. The situation is further improvedby stirring the dyebath to provide a good flow of solution across thefiber surfaces. This can be seen as eroding the boundary layer in which the negative potential is effective.

The Physical Dye-Fiber BondOne of the most poorly understood phenomena in dyeing is thehigh substantivity of direct dye anions and the anions of vat andsulfur dye leuco compounds for cellulose, and the causes of the physical dye-fiber bond. The anions from dissolved reactive dyesand azoic coupling components have less substantivity It has been suggested that for substantivity direct dye anions should be long and linear; but some substantive dyes are neither.Most direct and vat dye anions are essentially planar and rather large. Hydrogen bonding has been suggested as the source of the attraction, but there is evidence that cellulose is so strongly hydrogen bonded to water that preferential hydrogen bonding to theamino, hydroxy or other groups present within direct dyes seemsquite improbable.The most likely cause of substantivity appears to be the combina-tion of relatively weak forces, including the same short range vander Waals forces which contribute to the association of dye-ions with one another in the form of aggregates, either in solution or after the single dye-ions have passed across the potential barrier and have reached the fiber-water surface boundary Such tendencies to ag-gregate have been mentioned earliel; and certainly would be en-hanced by some combination of planarity length and linearity of dye anions, all of which favor such dye-ion to dye-ion association. It has been suggested that for substantivity direct dye anion sshould be long and linear; but some substantive dyes are neither.Most direct and vat dye anions are essentially planar and ratherlarge. Hydrogen bonding has been suggested as the source of the attraction, but there is evidence that cellulose is so strongly hydrogen bonded to water that preferential hydrogen bonding to the amino, hydroxy or other groups present within direct dyes seems quite improbable. The most likely cause of substantivity appears to be the combina-tion of relatively weak forces, including the same short range vander Waals forces which contribute to the association of dye-ions withone another in the form of aggregates, either in solution or after the single dye-ions have passed across the potential barrier and have reached the fiber-water surface boundary Such tendencies to ag-gregate have been mentioned earliel; and certainly would be enhanced by some combination of planarity length and linearity ofdye anions, all of which favor such dye-ion to dye-ion association.

Diffusion Into the FiberOnce the dye molecule, ion or other color bearing species (let us callit simply dye) has become associated with the fiber surfaces, thenext stage in the dyeing process is the diffusion of the dye from thesurfaces and into the fiber. This diffusion of dye into the fiber is acomplex subject to treat mathematically and much work has beenconfined to studying the simpler cases of polymer films and indi-vidual fibers with more or less circular cross sections. Even then,experimental studies of dye diffusion are potentially difficult.For a succinct discussion of the relationship between the dye dif-fusion coefficient, D, temperature and the denier of the fibers, seethe Dyeing Primer. The diffusion coefficient varies with, and can besensitive to, increases in temperature. It can be shown that practicaldyeing rates for cylindrical fibers of known radius, r, are related to the fractions D/ri and D / denier. What dyers need to be aware of is that when the dye has reached the surface of the fiber it has not finished its journey The dyeing process is not over until dye has diffused from the surface and to-wards the center of the fiber to give a more uniform distribution within the fiber. This situation can be summarized by means of a figure, which strictly speaking relates only to dyes of highSubstantivity applied at a relatively low percentage on the weightof the goods (%owg). Fig. 1.3 shows the concentration of dye at dif-ferent locations within the fiber. An early stage of dyeing is indi-cated in Fig 1.3., curve a, which shows a high concentration of dyeat the fiber-dyebath boundary [Dye]!, where the squared brackets indicate concentration and subscript f means in the fiber. There is a small tailing out of dye toward the center of the fiber some distance away As the dyeing process continues, curve b, the progression of dye into the fiber continues, the larger concentration of dye at thesurface diminishes and in the final stages of dyeing, Fig. 1.3, curvec, the distribution of dye throughout the fiber is essentially uniform.The relative velocity of surface sorption of dye and its subsequent diffusion into the fiber will determine the actual shapes of the ra-dial concentration profiles for the dye.

The rate of this diffusion process within the fiber might be quite slow and will vary widely from dye to dye. There is no agitation within the dyebath which can possibly affect the rate of diffusioninside the fiber which for all practical purposes behaves as if it werea flexible, solid rod. Dyeing, as far as exhaustion of the dyebath a flexible, solid rod. Dyeing, as far as exhaustion of the dyeb at his concerned, may be over long before the dye is uniformlydistributed throughout the fiber; which is why there is some confusion between the terms the end of the dyeing and true dyeing equilibrium. For those who wish to stop the dyeing as soon as the dyebath isat its maximum exhaustion, notethere are some material draw backs to having the dye on the

outer surfaces of the fiber as opposed touniformly throughout the fiber. The first one relates to the fastness properties of the resulting dyeing. If the dye is on or near the fiber surface then the ease with which it is lost to any subsequent wet or abrasive processes is much increased. More surprising perhaps; a fiber, with a fixed amount of color distributed in it, looks darker incolor the more uniformly that color is distributed. Consequently those content to stop the dyeing when the dye is merely distributed just within the fiber surface, overlook the possibility of an increased color yield, which will result as the dye diffuses (albeit slowly) into the fiber to become more uniform. Another benefit of uniform distribution of dye is that the lightfastness of the resulting dyeing is greaterThe property of having a fiber with most of the dye distrib-uted at the outer surfaces is called ring dyeing. Ring dyeing has thedisadvantages mentioned, but it is quite common, particularly incontinuous dyeing. Do not confuse this with ring dyed yarns!Bearing in mind the behavior of dye solutions, the fiber-waterboundary and the nature of the dyefiber bond detailed above, twodifferent but complementary pictures of the overall dyeing processfor anions on cellulose will now be introduced. One represents direct dyeing graphically The other describes an arithmetical modelfor dyeing in general, which is applicable to cellulose dyeing. Their usefulness derives from their relative simplicity

ThermodynamicsEquilibrium; Extent of DyeingThese areplotted against the concentrations of dye left in the bath at that time, [Dye]. Subscript e refers to the end of the dyeing, or close to equi-librium. These points lie on a continuous, smooth curve covering avery wide range of dye concentrations. This continuous curve iscalled a sorption isotherm. In the case of dyes for cellulose theseisothenns curve gradually towards the x-axis, but never quite leveloff. This behavior describes what are known as Freundlich isotherms.The Freundlich Isotherm is governed by the empirical relation-ship:

Here the power n is a value of no theoretical significance (empiri-cal); n is less than 1 but greater than 0, and k is a proportionality constant.

Freundlich Sorption Isotherm There are several implications to such Freuncllich isotherms: The partition or substantivity coefficient, K, which equals the concentration of dye in the fiber at the end of the dyeing (grams pergram) divided by the concentration of the dye in the solution atthe end of the dyeing (grams per gram) can be represented by the fractions:

From the gradient of the dotted lines drawn on Fig. 1.4b it can beseen that the substantivity coefficient K is not a true constant and gradually falls off as the amount of dye added to the dyebath ini-tially (or left in the dyebathfinally) increases. The importance of Kwill be discussed later, but it is related to the percentage exhaustion of the dyebath, %E. The drop in exhaustion with increasing depth of shade is well known to dyers, who expect less exhaustion from dyeings of heavy depths. To counteract the loss of exhaustion, it isusual to alter the dyebath conditions by adding increasingly moreand more salt as the depth of shade increases, to drive more dye onto the fiber:

The Freundlich isotherrns reflect the loss of activity or chemicaleffectiveness, of the dye in solution with increasing concentra-tion, perhaps due to aggregation. The Freundlich isotherms can also reflect the fact that the fibersurface, on which the dye molecules are initially absorbed, is het-erogeneous. Some of the locations, which get filled first, are more interactive with dye than those which are filled later. They are usually indicative of nonionic, or relatively weak bond-ing possibilities between dye and fiber, which has already beenmentioned. As more and more dye is added to a dye bath, more and more dye will continue to go onto the fiber. There is, of course, a practical limit as to how much dye may be put on any fiber; and this is dictated by the cost effectiveness of the change in color obtained by increasing the dye bath concentration. Above this limit further dye additions merely cause a gradual loss of brightness andgradual change of hue accompanied by diminished wetfastness properties.

Note: Recently several authors 3 have fitted direct dyeing datato Langmuir isotherms rather than tothe traditional Freundlich isotherm. Neither isotherm is ideal. Thesimple Langmuir isotlierrn implies a saturation value for dye on thefiber, which single direct dyes do not appear to have, even though(within a practical range) isotherms for individual dyes in directdye mixtures do seem to fit well to Langmuirequations.The em-pirical Frenmdlich isotherm can be shown to be unsuitable for kinetic modeling of the dyeing process.

Substantivity and Partition Coefficients

By rearranging we can bring the two rate constants onto the same side of the equation, and they can be combined into a numeri-cal constant, K, provided the concentration of dye in both solutionand fiber are in the same units-e.g., grams per gram-in which case the units cancel out.

K is known as the substantivity or partition coefficient shown inFig. 1.4b, and is a measure of the extent to which dye prefers thefiber to the dye bath in the particular dyebath conditions.

Liquor Ratio and ExhaustionThe partition or substantivity coefficient, K, can be used as a simpletool to demonstrate the general nature of the relationship betweensubstantivity; liquorto-goods ratio, L, and percent exhaustion at theend of the dyeing, %E. The percent exhaustion at any time duringthe dyeing is the percent of dye initially present in the dyebath whichhas left the dyebath for the fiber. The percent exhaustion at the endof the dyeing is of obvious importance to a dyer since it reflects howefficiently dye has been transferred from the bath to the fiber. Per-centage exhaustion can be written:

The equation is useful in showing the effect of varying liquorratios upon exhaustion, and the general magnitude of K, or the ex-tent to which dye prefers the fiber over the dyebath.'5 For a particular dyeing, exhaustion might be 90%-i.e. %E=90,when dyed in the plant at a 10:1 liquor-to-goods ratio, L=l0. Putting these values in Eq. 1.12, K must be equal to 9for the equation tobalance. But, at a 25:1 liquor ratio-e.g., in the laboratory when K=90and L=25, %E=78.3. This is a substantial loss of color yield, and color literally down the drain. The model confirms what experienced dyehouses already know.Plant dyeings at lower liquor ratios give higher color yields than laboratory dyeings at higher liquor ratios. It is well known that thedye content of laboratory formulations may have to be reduced bythe plant to give the same shades as those dyed in the laboratory Tobring plant and laboratory dyeings of cellulosic fibers closer together,the laboratory may elect to use larger amounts of electrolyte (salt)in an attempt to parallel plant dyeings and achieve higher exhaus-tion levels. If the relationship between plant and laboratory dyeingsis well understood, color recipes or dye formulations from the labo-ratory reproduce well in the plant.To have a dyeing with 95% exhaustion at a liquor ratio of 10:1,requires that K=190. The substantivity coefficient had to more thandouble from 9to 19to improve the percentage exhaustion by 5%,from 90% to 95%.In the case of the substantivity of anionic dyes for cellulosic fiber,it is truly remarkable that forces so poorly understood can cause thedyes to prefer cellulose to water by such large margins.

Fick's Laws of DiffusionDiffusion is the mechanism by which components of a mixture are transported around the mixture by means of random molecular (Brownian) motion (cf. permeationthe ability of a diffusant to pass through a body - dependent on both the diffusion coefficient, D, and the solubility coefficient, S, ie, permeability coefficient, P = D.S). Flynn et al. cite Berthalot as postulating, at the beginning of the nineteenth century, that the flow of mass by diffusion (ie, the flux), across a plane, was proportional to the concentration gradient of the diffusant across that plane. In the mid-1800's, Fick introduced two differential equations that quantified the above statement for the case of transport through thin membranes. Fick's First Law states that the flux, J, of a component of concentration, C, across a membrane of unit area, in a predefined plane, is proportional to the concentration differential across that plane (see note), and is expressed by: Equation for Fick's First Law

Fick's Second Law states that the rate of change of concentration in a volume element of a membrane, within the diffusional field, is proportional to the rate of change of concentration gradient at that point in the field, as given by: Equation for Fick's Second Law where t = time.