34 DIFFUSION OF SORBED SUBSTANCES A CONTRIBUTION TO THE THEORY OF DIFFUSION OF SQWBED SUBSTANCES INTO AND OUT OF FIBRES BY BERTIL OLOFSSON Swedish Institute for Textile Research, Gothenburg, Sweden Received 30th July, 1953 Thc relation between the equilibrium sorption isotherm s = f ( c ) and the sorption- coupled diffusion for fibres has been discussed. For finite bath exact solutions of the diffusion equations are obtained if (I) the actual part of the isotherm (corresponding to the range of bath concentration) is approximated by s = Rc (R = constant), or if (11) the same part of the isotherm is substituted by a line s = Ac + A0 and another part relatcd to the range of fibre concentration is substituted by 3s/3c = const. (not dependent upon A). The validity of these relations is tested on HBrlwool sorption curves : a satisfying constancy of the diffusion constant for a certain experiment is found and the difference between absorption and desorption values is cxplained by theory 11.Further applica- tions of these theories are summarized, viz. washing out acids from wool fabrics, diffusion of dyestuffs into fibres and moisture sorption on fibres. When discussing exchange of substances between a liquid (water) bath and fibres we must first consider: (i) the configuration of the fibre sample (yarn, fabric) that determines if the single fibres or only " macrosurfaces " of the sample are freely accessible to the liquid ; (ii) the circulation of liquid between the contact region of the sample and the rest of the bath. Some experimental work has been done to clarify the problems involved but these are very complicated. Furthcr clarification is obtained from this work too.In the theories here developed we make the assumptions: (i) the singleBERTIL OLOPSSON 35 fibres are in direct contact with the liquid (the equilibrium water-fibre phase established at zero time); (ii) the circulation of the liquid is such that a change in concentration just outside the fibre surface will at once be distributed throughout the bath. TmoRY.-lf the fibre is an absorbent for the diffusing material we could define a “ free ” concentration c as well as a “ sorbed ” concentration s. Generally c and s are functions of place, radial co-ordinate r, and time t. However, we further assume that there is an immediate equilibrium established between s and c at a point (r, t ) and this equilibrium follows the relation s =f(c).The diffusion is governed by a diffusion constant D which is generally a function of c, of the special phases (layers) of the fibres, of the direction of sorption, etc. Since the radial diffusion is generaIly the dominant process we further assume a homogeneous fibre and a D-value that is a constant for the solute and fibre considered. We focus our attention on the appropriate manner of applying a relation s ==f(c) for sorption problems. The fundamental mathematics are given by Wilson 1 on diffusion coupled with absorption into a homogeneous cylinder from a finite bath. The diffusion equation is written as We assume the fibres (radius = a, cross-section rra2) to be distributed in a bath of cross-section = A (cross-section of fibres excluded).Thus A/nd represents bath volume/fibre volume. Initially the diffusing substance has a concentration cg in the bath and CO everywhere in the fibre phase. Thus the total amount of substance at t = 0 will be rra2Co + Aco = MO -1 Aco. CASE I.-We make the assumption used by Wilson 1 that s = f(c) has every- where the form We further substitute s = Rc (R = constant). (2) y = 2n cr’dr’ (0 < r’ < a) s: in eqn. (1) together with relation (2) and obtain (3) There are two boundary conditions. The total amount diffusing substance is constant or 2.rr J: (c -I- s)r‘dr‘ + ~c =z M~ + A C ~ , or using (2) and (3) Further we have the initial condition for amount of substance in the fibre (6) phase ((2) and (3) are used), therefore We make the substitution (R + l)y(r, 0) = rrr2Co = r2Mo/a2 (0 < r < a, t = 0).36 IIIFFUSION OF SORBED SUBSTANCES in (4), (5) and (6) giving The solution of (8) has the form f == 2 BnrJl(qnr/a) exp (- P ~ O .(1 1) n Here and in following formulae Jn(x) denotes the Bessel function of order n for the variable x . By substitution in (8) we find Pn Z= Pqn2, (12) whcre /3 = D/(R 4- I)$. (13) Substitution of (11) in (9) gives aqnJo(9n) + Wl(Yn) = 0, a = A/[na2(R + l)], whcre and substitution in (10) gives where and Mt is defined as total amount of diffusing substance in the fibre at time 1. From (2) and (3) we find and further when (7), (15) and (18) are used, MI = (R -I- l)(a, t ) + Aco(1 + .)/(1 +a), (20) in (20), f(a, t ) is substituted using (ll), (14), (15), (16) and (17) to give From (21) we find For absorption wc divide (21) by (22) getting When MO = 0 wc get from (18), K = 0, and thusRERTIL OLOFSSON 37 For desorption we divide (21) by (18) giving When co == 0 we gct from (18), K cc, and thus On comparing the “ pure absorption ” (24) and “ pure desorption ” (26) wc obtain the relation CASE IT.-Assumption (2) is often a very rough approximation.For dye- stuffs with their high affinity, and also for acids on wool, we know, however, that $9 c, and we further put s -I- c = c, and Cr’dr’ (0 < r <a). (3’) Now we make the transformation suggested by Standing et al.2 and Vickcrstaff 3 for dyestuff diffusion using (2’) and (2”) : giving the new form of (l), If W is substituted from (3’) we get To get an exact solution, we further assume that there is a linear relation between s and c at the fibre surface, Then the boundary condition for constant total amount of diffusing substances will be (compare (5)) s == xc 4- A() (r = a). (2’”) =: (M” + Aco)(l -1- A) - AA”, (5’) (6’) (7’) (8’) r - a and the initial condition corresponding to (6) We now make the substitution H(r, 0) = .rn‘2Co = Mor2/a2 (0 < r < a, t = 0).(Aco + M”)(h -!- 1) + AA” H(r, t ) = in (4’), (5‘) and (6’) getting r2 + F(r, t ) (A -t 1 ) d -i- (A/T) (9’)38 DIFFUSION OF SORBED SUBSTANCES The solution of (8’) is F = ~ i r ~ l [qnr/al exp (-- p i t ) , N and substitution in (8’) gives Pn’ == p‘qn2, (13’) NqnJdqn) i- 2Jl(qn) 0, ( 14‘) where a = A/[nu2(A 4- l)].(1 5’) D 3c where p ’ = - - a2 as* Substitution in (9’) gives Substitution in (10’) gives and Mt is found from (2”) and (3’), KO = CO/(CO I- cox + ho). and further, if (7’), (15’) and (18‘) are used, 4 c o -1- cox -I- AO” -I- ( K O / 4 l (a -1- 1)(A -1- 1) _ _ _ _ _ ~ Mt = F(a, t ) -I- - * Now F(a, f) is substituted, using (ll‘), (14’), (15’), (16’), (17’), whence A(c0 + cox -t XO” + ( K 0 / 4 1 (a -I- -I- 1) For absorption, we divide (21’), by (22’1, Mw = When CO = 0 we get from (18’), KO = 0, and thus For desorption, we divide (21’) by (compare (18’) and (15‘)) Whcn (i) thcre is no substance in thc bath at t = 0 or co = 0, and (ii) ho = 0, we get from (18‘), /co = co, and therefore The relation between (24’) and (26‘) is given by (27).BERTIL OLOFSSON 39 For the relation between (25’) and (26’) we get 1 = (1 - ;) -’- KO’ CASE 111.-Experimental equilibrium sorption isotherms generally give thc relation C = c + s =f(c).If (2’) is valid, a good approximation for this expression is s = f(c). This relation is generally not linear and hence it is not possible to get an exact solution of the diffusion problem. However, we may use a method in which differentials are changed to finite diffcrences. For the absorption this method is given by Crank 4 and thc corresponding formulae for general sorption are easily deduced. The equations governing thc process are 0 3 3c 3s -- r 3r (r;) = -+ z, s =f(c), (29) and the boundary conditions are Ac.,,. -1- 2nl; (c -1- s)r’cir’ == Aco + n-azco ::- Aco -t- )do, (30) (3 1) Reduced variables are introduced, t is changed to T = Dt/n2 and r changed to p == r/a, the fibre cross-section is divided in circular shells of thickness dp and thc time is divided in intervals d7.The formulae now used are to be found in Crank’s paper.4 APPLICATION.-kI applying these theories to experimental results it is con- venient to use graphical representations. As we generally measure Mt as a function of t, the different equations for Mt/M0 or Mt/Mw should be represented. Now it is possible to find relations between equations such as (27) and (28’), and only one of these equations for casc I and case I1 must be plotted. We have chosen eqn. (26) or (26’). We take a and pt (or P’t) as parameters. For a given a-value we find values qn from (14), using tabulated values for the Besscl functions JO and J1.5 The a and the calculated qn values are substituted in (26), together with chosen Pt values in the exponents pnt = qn2Pt, and thc corresponding Mt/Mo values calculated.In the graph Mt/M0 is plotted against the transformed variable p - lOO/(l -t- a) and plots for a given ,&-value are constructed. The form of these diagrams is shown by fig. 1. C := CO ( t = 0, 0 < r < a). For Pt = co we find from (26), This shQws that the “ limit curve ” is a straight line from its origin to p = 100, Mt/Mo =: 1.000. For small values of t, expression (26) gives a very slow con- vergency of the series in exp (- pnt). Here we must use asymptotic expansions of the Bessel functions. The appropriate methods are given and discussed by Crank .6 SORPTION OF EIBr ON WOOL The validity of these theories has been tested by some careful measurements on the kinetics of HBr sorption on wool.The reason for using HBr was that this acid had been employed by Larsson and Lindberg (unpublished) in some earIier investigations of de- sorption using radioactive bromide ion. The experimental arrangement in the present case was designed by Lindberg.7 The experiments here reported have also been made by Lindberg, in connection with his work on thc reIations between acid diffusion constants and surface treatments of w00l.7~8 The fibres are ground in a Wiley mill and suspended in the bath, whcre they are kept in rapid movement by a stirrer. The change of concentration is measured by the change in electrical conductivity of the bath.A conductivity cell in the bath is part of a resistance40 DIFFUSION OF SORBED SUBSTANCES bridge with a comparison cell in a constant concentration bath and the bridge is coupled to a recorder, where the relative bath concentration is thus registered as a function of time. If now the initial amount of HBr in bath and fibres is known and also the constant volume of thc bath, we can easily calculate values of Mt (and Mt/Mo, or Mt/M,). The following experiments were madc. A 0.5 g sample of wool was ground, extracted with ether and alcohol, suspended in 5 ml distilled water and added to 250 ml 0015 M HBr in the conductivity vessel. The absorption curve a t equilibrium was recorded (A). Now the sample was filtered and the solution replaced by distilled water.The sample (together with a small amount of equilibrium solution from (A)) was added and the de- sorption curve a t equilibrium recordcd (B). Now the sample was carefully washed with distilled water and after this a new absorption curve was obtained (C), the HBr concentration being now 0.001 M, this figure corresponding to the final bath concentration in (B). All the experiments were made in duplicate. To calculate p values according to case I from fig. 1 we must first determine a or p using eqn. (32), e.g. withp = 100 M,/Mo for desorption. In “ pure absorption ” (Mo-0) FIG. 1.--Mt/M0 as a function of p = l00/(l -I- a) and @ x 10-3 (the figures given on the curves) from (12), (14) and (26). we calculate a from B,/M, (B, is the amount of acid in the bath at t = m ) using (22) : For such a p value and MJMo from experiments (or from cxpcrimental Mt/M, using (27)) fit and thus l3 values are determined from the graphs, For all the experiments A, B and C the /3 values show a satisfactory time constancy.For A and also B, f i decreases ; for C it increases somewhat with time as seen from fig. 2. We may calculate a simple number average a = ZP/n (n = number of values) or a time average = Zpt/Zr; this is probably more significant since every /3 value is calculated as a constant for the time 0 to t (table 1). A small deviation of %/a from 1 denotes a good time constancy of 8. However, is very different in the three experiments A, B and C. From (13) we might suppose that thc reason for this is the variation of X.According to (2), R should be some sort of average of the quotient SIC within the concentration region for the actual cxpcri- ment. Now we can determine R from thc cz value using (15). For this we must determine A/m2 = bath volume/fibre volumc. If the dry fibre weight is g with the density p, the dry volume will be g/p and th:: swollen volume (g:p)(l -1- s), where s is the fractional volume swelling. Substitution in (15) gives R. We find (table 1) important differences in cases A, B aiid C . Howcver, if these values are substituted in (13) we get values of D/a* = B(R -t- I ) . From table 1 we find that A As p N 1.3 and s ci 0-3 we get nu2 =< g.BERTIL OLOFSSON 41 10 20 t(rnrn) 30 4 0 FIG. 2--15 (case I) and -15’ (case 11) as a function of t for HBr/wool sorption curves A, B and C .TABLE AD ADSORPTION AND DESORPTION OF HBr ON WOOL 0.5 g dry wool -1- 255 ml bath (A/,& 510) absorption A desorption B absorption C - sample sample sample rcfercncc -~ ~ _ _ _ _ _ _ _ 1 2 range of c mmole/ml 0.0147 --> 0.0136 range of s mmole/g wool 0.0 -+ 0.616 ci 13.5 12.5 - 8n 0.0546 0.0492 81 0.0516 0.0480 R 36-7 39-9 (min-1) 2.02 2.01 (D/a2)r (min-1) 1.95 1.96 R (average) 38.3 h 14.9 A0 0.3 18 a 322 P’n 0.0592 0.0521 F t 0.0572 0.0505 @’n 0.88 0.78 @’1 0.85 0.75 c - 3s/3c 21.2 (D/a’)n (min-1) 1.26 1 * 1 1 (D/a2)r (min-1) 1.21 1-07 1 2 0.0 -> 0.00083 0,615 -+ 0.182 2.7 2-1 0.0230 0.0242 00199 0.0238 187.3 237.3 4.32 5.76 3.74 5.66 212.3 169.4 0.009 3.0 0.0229 0.0204 0.020 1 0.0 193 3.88 3.46 3.41 3.27 14.6 0.33 0.30 0.29 0.28 1 2 040098 -> 0.00075 0.0 -+ 0.135 3.3 3.1 0.0123 0.01 13 0.0128 00119 152.5 161.7 1.89 1.84 1 -97 1-94 157.1 89-8 0.058 5.6 0.0140 0.0131 0.0142 0.0135 1-26 1-18 1 -28 1-21 160.9 2-25 2.1 1 2.29 2-17 (32) and (32‘) fig.1 and 2 (1 5 ) (1 3) (1 3) “ least square ” (1 5’) fig. 1 and 2 ‘‘ least square ” (1 3’) (13’)42 DIFFUSION OF SORBED SUBSTANCES and C give the same L)/cr?- value but B a D/& value about 3 times as large. This resuIt is probably a result of the failure of the theory of case I. We can also compare our R values from (15) with values of s/c from acid/wool titration curves, giving s in units of mmole/g dry wool as a function of pH. Thus s is determined using the same assumptions as for R from (15), (1 g dry wool 21 1 cm3 swollen wool), e.g.all the fibre-water phase is assumed to be accessible for diffusing ions to the same extent. This is not in agreement with the “ Donnan theory concentration ”, where only the swelling water is taken as a solvent medium for intcrnal ions,g but it is a rather practical assumption. c = aH/fH is calculated from pH = - l o g l ~ a ~ and fH values tabulated. The s =: f ( c ) curve from Steinhardt-Harris HCl + wool titrations at 25” ClO is drawn in fig. 3. On the same graph we draw the s = Rc curves from R values (table 1). We now com- pare the position of these lines with the corresponding regions of variation for c and s on the experimental isotherm (fig. 3 and table 1). In case C, the R line corresponds rather well to the isotherm if the region of c as well as the region of s is considered.In case A, R corresponds well to the small c region and also gives an acceptable average for the s region, although this region is very large and thus the curvature large and the FIG. 3.-Relations between experimental s = f ( c ) curve and approximations used for HBr + wool sorption curves A, B and C . line a bad approximation. In case B, R corresponds well to the small c region, but is wholly outside the large s region. From these considerations and the good time con- stancy of /3 we might conclude that the appropriate R value in case I shouId be taken from the linear approximation of the isotherm between the experimental limits of bath concenfraf ion. We now apply the theory of case 11. Here the relation (2”’) is valid only for the fibre surface and hence this necessitates a linear substitution for the isotherm in the small bath concentration region.We have used the method of least squares for the corresponding parts of the isotherm (fig. 3) to get the A and A0 values. a is now obtained from (15’), where A/rra2 is calculated as before. For absorption, /3’ values are calculated as before from fig. 1, using (27), (12’) and (24’). For desorption, (M/&bf~),~ = o3 to be used in the graphs is calculated from (28’), where KO = Co/Ao according to (18’), (CO = 0). As seen from fig. 2 and table 1 the j3’ values are not very different from the /3 values for case I. Of course the method here given to calculate A does not fit the theoretical curvc at t = co . This is really reflected in an increasing t h e dependence of /3’ at large t values.Of course we might also fit the A-value to the experimental t = co conditions, e.g. put A = R for case I, or make a compromise between this condition and the “ least square ” condition. We find that although this region for the A values is rather large (R -> A in table l), theBERTII- OLOFSSON 43 corresponding region of p’ values is small @ -+ in table 1). However, the real diffusion constant in case I1 is found from (13’) : D/a2 = /3’ 3s/3c. A constant value of #I‘ means a constant value of 3c/3s, e.g. we also here make a linear substitution. But the important thing is that 3c/3s is not necessarily related to A. 3c/3s should be some approximation for the region of concentration within the fibre. If we put 3c/3s = 1/A we get the same kind of differences between D/a2 in cases A, B and C as if dc/bs = 1/R, or = 1/(R + l), case I.Now we make the assumption that 3c/3s is governed by the region of change for s in the fibre. We thus calculate 3c/3s from the isotherm (fig. 3) by the least square method for the s regions in the three cases; we have roughly (table 1): A region = B region + C region. Table 1 gives the D/& values obtained from (13’). We also here find large differences in the three cases, but the im- portant fact is that D/a2 for desorption (B) is much smaller than for absorption (A and C), contrary to the previous results. Thus it is probably corrcct to attribute differences in D/a2 to the choice of R or We obtain the following conclusion. Theory 1 is generally applicable for experiments where the change in bath conccntration is not very large.When comparing diffusion constants theory I1 should be used. For absorption, how- ever, thcory I is also now applicable as the parameters h and 3c/3s in I1 may both be substituted by the same value for R in I. For desorption theory, I1 must be used, i.e. h is calculated just as R for I, but 3c/3s is taken as another average for the fibre concentration range. For our case B, we assume D/a2 = 1.95 (= the absorption value), thus 3c/3s = /3’/1*95 = 0-0227/1-95 (from table 1) = 1185.8. The value 85.8 is smaller than the R and A value in B but larger than the 3s/3c value (table 1). However, for assessing thesc conclusions calculations according to case I11 must be made. The relation (29) to be used is (3 3) deduced from the Gilbert-Ridcal theory.9 Here R’ is taken from the equilibrium condition (t = m) for the experiment considered and s, is put equal to 0.86.9 The result of these calculations will be given later. WASHING PROCEDURES.-TheSe considerations have found an interesting and useful application for analysis of washing processes.11 We have made experi- mental investigations of the kinetics of winch-machine washing of sulphuric acid from wool fabrics, using baths of pure water as well as of neutralizing agents. For water-washing with changes of water the desorption theory of case I is used. If there is a continuous in- and outflow of water during the experiment we have used the idea of an “ expanding bath volume ” for the theoretical interpretation. When neutralizing agent is present in an amount equivalent to the total amount of acid, wc have considcred A / r G in (15) to be infinite, e.g.a = 00 or p = 0. Howevcr, in such cases thcre is generally a “ sorption level ” left in the material, its “ height ” varying with the character and amount of neutralizing agent. If v. and p is determined from this level, then p > 0 gives a very good timc constancy of /3 (from fig. 1). The explanation of this “ fast level ” might be that on the “ small sorption part ” of the sorption isotherm (fig. 3) R is increasing towards infinity at the same rate as is A/ra2; a thus is finitc. We may consider the points (i) and (ii) of the introduction. Probably condition (ii) is satisfied in this washing procedure but not condition (i).The failure of condition (ii) should be reflected in bad time constancy of p (Larsson, Tenfalt, unpublished) which is not observed here. The failure of condition (i) means, firstly, a decrease of the diffusion rate,l2 which is observed here, being much smaller than for free fibres. With neutralizing agents, however, the contact between liquid and single fibres seems very satisfying, being of the magnitude of for free fibres. Now from thc thcory we haw deduced relations between bath ratio, tempera- ture of bath, time of washing and number of washings. Also the cost of washing may be considered as a function of thesc factors and thus it is possible to determine a combination of thcse factors that gives minimum costs. values. s/(S, - S) = R‘c44 DIFFUSION O F SORBED SUBSTANCES DIFFUSION OF DYESTUFFS.-A micro method for studies of dye absorption has becn developed.13 Singlc fibres are dyed under fixed conditions in a spccial apparatus ; after fixed intervals of time, these are immediately frozcn with carbon dioxidc and dried in Ihe frozen statc.Cross-sections are cut with a microtomc, microphotographs taken and the distribution of dye recorded with a photonictcr. Thus we obtain values of the total concentration C(r, t), or rathcr C(r, t)/C(a, 0), where C(a, 0) is the concentration at thc fibre surface (not outside thc fibre) initially. We apply the theory of case 11 and from (3’) obtain 3H/3r is calculated from (7’) and (11’) and after simplifications we get If the fibre is initially free from dye, we get KO = 0, (18’), and if we assume an in- finite bath, we get a = co, (15‘), the resulting equation being Now experiments with viscose model fibres and direct dyestuffs give two character- istic results : (i) the absorption of dye in the external layer is very rapid but the further diffusion inwards rather slow ; (ii) the dye moves inwards with a very sharp boundary.(i) may be explained by the large “ free concentration ” at the fibre surface. (ii) is possibly interpreted as a decrease of Ddc/?s when passing the boundary inwards. This means a decrease of 3c/3s, e.g. there is a decreasing relative amount of dyestuff free to diffuse, and this wiIl sharpen the boundary, increasing the speed of the back and decreasing the speed of the front dye particles.However, the experimental method must be refined for further investigations. DIFFUSION OF MOISTURE.-A~ apparatus for determining the kinetics of vapour sorption from the change in capacity of a cell with loosely-packed fibres has been constructed and absorption and desorption curves for wool at different humidities for a given velocity of air current studied (unpublished work). The theories have been applied (a = co). It is found that /3 changes significantly during one experi- ment and is also a function of the humidity limits used and of the direction of sorption (absorption or desorption). It is evident that the theory must be de- veloped in several respects. Thus the validity of the linear approximation of s ~7 f(c) should bc investigated. It is also most important to considcr the swelling effects and their hysteresis. The influence of the temperature gradient from sorption heat must also be examined. An important critical study by Crank and Park 14 discusses other facts about such sorption proccsscs. I thank Dr. J. Lindberg very much for putting his expcrimental results at my disposal and furthermore for stimulating work and discussions on the kinetics of “ textile wet processing ”. I am very grateful to Prof. N. GralCn for his active and encouraging interest in these problems. Technical assistance and help with calculations has been given by several persons at the Institute to whom I express my gratitude. 1 Wilson, Phil. Mag., 1948, 39, 48. 2 Standing, Wanvicker and Willis, J. Text. Inst., 1947, 38, T335. 3 Vickerstaff, The Physical Chemistry qf Dyeing (Oliver and Boyd, London), p. 123. 4 Crank, Phil. Mag., 1948, 39, 140. 5 Watson, Theory of Bessel Functions (University Press, Cambridge), p. 666,BERT I L 0 L 0 F S SO N 45 6 Crank, Phil. Mag., 1948, 39, 362. 7 Lindberg, Text. Res. J. (to be published). 8 Lindberg, Text. Res. J., 1950, 20, 381. 9 Olofsson, J. SOC. Dyers Col., 1952, 68, 506. 10 Steinhardt and Harris, J. Res. Nat. Bur. Stand., 1940, 24, 352. 11 Olofsson, Medd. Sv. Textilforskirgsinst., 1953, nr 34. 12 Olerup and Lindberg, J. Sac. Dyers Col., 1950, 66, 148. 13 Olofsson, Mecld. Sv. Textilforskningsinst., 1953, nr. 29. 14 Crank and Park, Trans. Faraday SOL, 1951,47, 1072.