Joumal of Crystal Growth 92 (1988) 61-68 North-Holland, Amsterdam MASS-TRANSFER AND SUPERSATURATION IN CRYSTAL GROWTH IN GELS Application to CaSO, 2H ,O M. PRIETO, C. VIEDMA, V. LÓPEZ-ACEVEDO, J.L. MARTÍN-VIVALDI and S. LÓPEZ-ANDRÉS Departamento de Cristalografia y Mineralogia, Facultad de C.C. Geolbgicas, Universidad Complutense, Madrid, Spain Received 22 January 1988; manuscript received in final,form 30 May 1988 Supersaturation evaluation is an essential requirement to describe, confront and explain crystal growth experiences. However, in the particular case of crystal growth in gels, experiences are often described by attending to the initial concentration of reagents. This fact is wnnected with deficiencies in the theoretical quantification of mass-transfer, and therefore in both time and location prediction for the first precibitate. In this paper laboratory experiences have been specifically designed to test supersaturation evolution through an actual (finite) diffusion system. The problem is carried out by keeping into account several complexity factors: free ions as well as complexes and silica gel Na+ and C1- "unloading" are considered to evaluate the supersaturation. 1. Introduction Supersaturation evaluation is an essential re- quirement to describe, compare and interpret crystal growth experiences. In the case of crystal growth from solution, the relative supersaturation may considerably depend on the applied con- centration definition [l]. The use of concepts such as "thermodynarnic supersaturation" [2] and "growth affinity" [1,3] is an attempt to get rigor- ous and comparable expressions for the driving force. Despite all the afore-mentioned facts, in crystal growth in gels the experiments are often described by attending to the initial concentrations of the diffusing reagents and to the geometric configura- tion of the experimental system. Any quantitative mention of supersaturation is absent. This fact is related to theoretical deficiencies about two aspects of the technique: the mass-transfer quantification and the establishement of rigorous crystallization criteria to predict time and location for the first precipitate. In crystal growth in gels, volume mass-transfer occurs by diffusion. Consequently, theoretical treatments have usually been based on Fick's dif- fusion law analytic solutions. However, in such analytic approach to the mass-transfer problem, diffusion systems and source reservoirs are respec- tively considered semi-infinite and inexhaustible, whereas actual systems are finite. Recently, Henisch and García-Ruiz [4] have described a microcomputer implementation whch allows one to solve Fick's diffusion equation, as a time function, by numencal methods. This treat- ment is a suitable procedure for finite systems with differents configurations. Regarding the second theoretical problem, the crystallization cnteria, Henisch and García-Ruiz [5] consider the case of a system where two coun- ter-diffusing reagents meet to generate a sparingly soluble reaction product. Following these authors, for a precipitate to form, it is not enough that the local concentration product exceeds a critical value K S . A new concept, "equality range" of reagent concentrations, stated in equivalents, needs to be introduced in order to explain experimental re- sults. By using both conditions, Henisch [6] carries out a microcomputer analysis to predict the first and subsequent precipitates position, in the case of Liesegang ring formation. Unfortunately, up to now, such microcomputer implementations are based on a one-dimensional . algorithrn and do not distinguish between micro- and macrocrystalline precipitates. Morwver, the systems they have worked with are hypothetical 0022-0248/88/$03.50 O Elsevier Science Publishers B.V (North-Holland Physics Publishing Division) aespino Lápiz 62 M. Prieto el al. / Mass-lransfer and supersaturation in ctystal g~owth in gels and several assumptions have been made for the sake of simplicity (complete ionization, the diffu- sion constant is independent of concentration levels, etc.). So, this approach is a good instrument in an interpretative way, but not in a quantitative one. This paper deals with laboratory experiments which have been specifically designed to test su- persaturation evolution (space and time). This al- lows one to overcome the problem by attending to actual complexity factors, and to check the effec- tive weight of the comrnonly made assumptions in the theoretical approach. So, the influence of con- centration levels in the diffusion constant, and the presence of both, free ions and complexes, in solution are considered. 2. Experimental Mass-transfer testing was carried out in a dou- ble diffusion-reaction system (fig. 1). In such a system, two reagents, CaCI, and Na2S04, diffuse through a silica gel colurnn (9 mm diameter and 280 mm long) to meet and generate Caso4. 2H20 as a reaction product. Silica gel was prepared by the acidification of a Na2Si0, solution (Merck, sp.gr.: 1.059 g/cm3; pH 11.2) with HC1 (1N) solution until the desired pH leve1 (5.5 in al1 the experiences) was obtained. The source reservoirs were filled with 8 cm3 of CaCI, and Na2S0, solutions (initially 1 M, 0.5 M and 0.3 M). The temperature of the experiments was 25 O C. Mass-transfer testing was carried out by re- actant diffusion during previously stated periods * GEL CDLUYN c Fig. 1. Gel growth system schematic representation. of time (3, 8, 16, 24, 32, 40, 48, 56, 64, 72 and 80 days). After the diffusion period, the gel colurnns were sliced, and the gel slices (10 mm long) were chemically analyzed. Chernical analysis of calcium and sodium was done by atomic absorption spectrophotometry (an UNICAM SP90 atomic absorption spectropho- tometer with acetylene burner was used). The de- tection limit was 0.0001 mmol/l for calcium and 0.0004 mmol/l for sodium. The SO:- concentra- tion was analyzed by precipitating the sulphate with a barium chloride standard solution in acid medium. Then the barium excess was measured by atomic absorption spectrophotometry. So, preci- sion for sulphate analysis was 1.04 mmol/l. How- ever, the accuracy lirnit is only determined by the analytical method for low wncentration values. Effective control of chernical determinations is done by the slicing process precision: a gel colurnn 5 is easily deformable and consequently an error up to 1% for gel slice weight may be expected. So, the error margin for concentration values is 1% con- centration dependent. Chemical constituent concentrations, measured in that particular way, are related to the whole volume of the gel slice (0.65 cm3 = polymers + interstitiai solution). Consequently, these values may be corrected to obtain the actual concentra- tion of interstitial solutions. In order to establish d the actual concentration values, the "effective water" per unit of gel volume had to be de- termined. With this aim, 100 cm3 of 1M solution and 100 cm3 of inert gel were put in contact with each other in a crystallizer. The interface between gel and solution was a circle of 10 cm of diameter. After 90 days, the concentration of the solution was measured, and out of this datum the weight percent of "effective water" was found; in this case 95.6%. This value is near the weight percent of total water in the gel, which reflects the weak polymer-water interaction in these kinds of gels. The so-corrected concentration values are the total values for each entire slice, by removing the concentration gradient inside each slice. SO, the concentration graphics throughout the gel colurnn are stepped, on the understanding that the actual concentration gradient would be done by the step enveloping. M. Prieto et al. / Mms-transfer and supersaturation in crystal growth in gels 63 3. Mass-transfer in finite diffusion systems In finite systems, mass-transfer follows a differ- ent outiine from the thwretical forecast for semi- infinite systems. Total concentration values of calcium (mrnol/l) through the diffusion column are represented in fig. 2a and pointed out in table 1. The siices are numbered from 1 to 28, the first being the nearest one to the CaC1, mother solu- tion. Different curves wrrespond to different dif- fusion times. Curve morphologies continuously change in time. The curve corresponding to a short diffusion period (8 days) is the only one that shows an aspect similar to that expected, from a thwretical point of view, for a semi-infinite system with infinite solution reservoirs. This curve also agrees with the one obtained by Henisch and García-Ruiz [4] for brief computation times. For longer diffu- sion times, the curves are not comparable with those obtained by these authors, since, in the experimental system, the boundary conditions are different (for instance the end of the gel column (slice 28) is not a sink of concentration O). The concentration decreases in the initial ex- treme, whereas it increases in the final extreme as diffusion time passes by. This fact means a con- tinuous change in the edge conditions that pre- cludes the analytic treatment of the mass-transfer. If the formation of a precipitate would not inter- rupt the process, concentration curves would logi- cally tend to be straight horizontal lines that would represent the final concentration, homogeneous for the entire gel column. In fig. 2b evolution in time of the concentration is drawn for severa1 gel siices. In al1 cases, after an initial upward fiight, the concentration decreases. The siice 1 is not an exception but its upward fiight occurs during the first days of diffusion and therefore the increase is not shown in the graphc. Only the furthermost slices show a continuous concentration increase during the diffusion peri- ods concerned. This is why al1 curves converge towards closer and closer values as diffusion time increases. As complementary information, the total con- centration values of sulphate throughout the gel column are pointed out in table 2. The pH profiles are also given in table 3. (Complete mass-transfer data used for this work are available to readers upon request.) 4. Gel Na+ and CI - unloading During the preparation of the described silica gel, NaCl is formed as soluble byproduct that is incorporated to the interstitial solution. So the GEL SLICES DIFFUSION TIME IDAYSI CALCIUM CALCIUM Fig. 2. Transport of caldium Mother solution: 1M CaCI,. (a) Total calcium concentration throughout the gel column for different , diffusion times. (b) Calcium concentration evolution in time for different gel slices. mMl l mMI a 1 b( 64 M. Prieto et al. / Mas-transjer and 'supersaturation in crysial growfh in gels Table 1 Totai caicium concentration (mmol/l) throughout the gel col- umn for different diffusion times; mother solution 1M Ca,CI Siice Diffusion time (days) number 16 32 64 80 expression "inert gel technique", when it is ap- plied to experimental devices as those used here, it is only a conventional term but not an actual one. In fact, the presence of C1- and Na+ in the interstitial solution means, in the first place, an alteration of the reagent's diffusion coefficients, and second, that the interaction of these ions with the reagents in solution may produce complex formation. For instance, the formation of the NaSOT ionic pair is very important in the growth of gypsum crystals, because it involves an im- portant concentration decrease of SO:- free ions, so that the actual supersaturation of the system is smaller than in a Na' free gel. The interference of interstitial Na+ and Cl- in mass-transfer and supersaturation may not be re- moved in a rigorous treatment of the problem. On the other hand, the initial homogeneous con- centration for both elements changes with diffu- sion time: the two solution reservoirs are, at the same time, reception chambers for C1- and Na+ ions coming from the gel which is " unloaded" this way. In fig. 3a, gel Naf unloading is shown. The reservoirs are initially filled with 8 cm3 of bidis- tilled water. Once again, both edge conditions and curve morphologies change continuously with dif- fusion time. The curves would tend towards the horizontal at the limit. Fig. 3b shows the interaction influence between different species in solution in the diffusivity of each one. In this case the solution reservoir No. 1 was fiíled with 1M CaC12 and No. 2 with bidis- tilled water. Consequently the gel unloading be- comes asyrnrnetric, decreasing the sodium diffusiv- ity towards the reservoir that was filled with the reagent. 5. Free ions and ionic associations throughout the gel column For gypsum growth [7], the supersaturation can be given as follows: where a(Ca2+) and SO:-) are the activities of the free ions in solution and K,, is the thermody- namic solubility product (K,, = 10-4.62 at 25O C). Eq. (1) shows that the estimation of super- saturation requires knowledge of the free ion ac- tivities. Van Leeuwen and Blomen [3] express the driving force in terms of mole fractions and activ- ity coefficients from the Debye-Hückel theory for moderately concentrated solutions. These authors have shown that this method gives accurate results up to very high supersaturations, even for the fairly soluble gypsum. The application of the Debye-Hückel formula for an estimation of the activity coefficients requires a calculation of the ionic strength of the solution, and this involves the consideration of the ionic associations as well as of the existing free ions in solution [8]. This is especially important in M. Prieto et al. / Mms-transfer and supersaturation in ctystal growth in gels 65 Table 2 Total sulphate concentration (mM/l) throughout the gel column for different diffusion times; mother solution 1M Na, SO, Slice Diffusion time (days) number 8 16 32 64 80 1 677 649 594 476 406 2 516 555 575 47 3 407 3 380 486 554 467 405 4 262 364 511 462 406 5 161 305 466 459 404 6 113 260 406 449 400 7 54 184 294 427 386 8 26 101 230 392 374 9 19 62 181 346 356 10 13 41 117 287 334 11 11 23 61 217 310 12 9 17 42 181 281 13 7 13 36 169 262 14 4 1 10 28 143 238 15 3 ' 8 20 119 213 16 2 6 15 100 192 17 1 3 11 84 168 18 1 2 9 70 145 19 - 2 6 58 120 20 - 2 5 49 91 21 - 1 3 42 78 22 1 2 33 67 23 - 1 2 28 57 24 - - 1 23 50 25 - - 1 20 43 26 - - 1 17 39 27 - - 1 14 34 28 - - 1 11 29 the case of gypsum, since the CaSO, soluble com- plex is almost as abundant in solution as the free ions [7]. In this particular way, based on the analytic values of total concentrations of chemical con- Table 3 Profiles of pH for different diffusion times; mother solutions: 1M CaC1, and 1M NA,SO,; initial gel pH: 5.5 Slice Diffusion time (days) number 8 16 32 64 80 stituents in solution, free ion and complex con- centrations must be calculated. This involves the resolution of a equation system that is formed by mass-balance equations and by ionic association equilibna. In the present case, the total wncentra- tions in calcium, sulphate, sodium and chlorine are known from the chemical analysis of gel slices, the pH being a complementary piece of informa- tion. From these data, the concentration of both, free ions ([Ca2+ 1, [SO,Z.], [Cl-1, [Na-1, [H+]) and soluble complexes ([CaSO,], [HSO;], [CaOHf], [NaSO;], [NaOH]), must be figured out. With this aim, a microcomputer programme was used. The programme is similar to the one described by Lundager Madsen [8] for calcium oxalate. The. equilibnum constants of ionic associations were obtained from Martell and Srnith [9]. Fig. 4a shows the total concentration in calcium and sulphate ([ca2+lT; SO:-]^) of the gel slices. M. Prieto et al. / Mass-transfer and supersaturaiion in ctystal growth in gels GEL SLICES G E L S L I C E S Fig. 3. Gel Na+ unloading. (a) Both solution reservoirs filied with bidistilled H,O. (b) Solution reservoir No. 1 filled with 1 M CaCI , . These data are compared with the values of their respective free ions. The curves have been ob- tained from 1M mother solutions after 50 days diffusion time. Total concentration values are much higher. This is due to the existente of calcium and sulphate forming complexes in solution. Fig:4b shows the concentrations of the most important wmplexes with SO:- as a constituent. The concentration of the soluble complex CaSO,, the major calcium "consumer", is maximum in slices 14 and 15. That is the point where the product of calcium and sulphate total concentra- tions is higher.On the contrary, [NaSO;] is higher where [SO:-] is also higher, that is to say, near the solution reservoir of this reagent. So, whereas the greatest difference between [Cal, and [Ca2+] is Gel Slicee Gel Slicee Fig. 4. Free ions and complexes throughout the gel wlumn. Mother solutions: 1M CaCl, and 1M NazS04. Diffusion time: 50 days. (a) Total concentrations in relation to the respective free ion concentrations. (b) Concentration of sulphate main species throughout the gel column. M. Prieto et al. / Mas-transfer and supersaturation in crystal growth in gels found in slices 14 and 15, in the case of sulphate this difference is maximum for slices 24-28. The consideration of ionic associations is very important to calculate the actual supersaturation, considerably lower than the values that would be expected from the total concentrations. Further- more, the unequal distribution of the different ionic associations throughout the gel column dis- turbs the spatial distribution of supersaturations, which has undoubted implications in the first pre- cipitate location. The knowledge of the concentration of free ions and complexes throughout the diffusion col- umn allows us to work out the ionic strength and the activity coefficients r of the free ions. Finally, the knowledge of [ ~ a / + ] , [SO:-], r ( c a z + ) and SO:-) allows us to calculate a(caz+) and a(s0:-) and hence the actual supersaturation B. 6. Supersaturation evolution throughout the gel c o l m It has usually been considered that the first precipitate may develop at the point where the concentration product is maximum. Therefore, if the reagents' diffusion coefficients are equal (com- plete ionizations and independence of the diffu- sion coefficient related to the concentration leve1 are assumed in this asserveration), the first pre- -- so Days 64 - . . . . . . . 50 - ----. 40 - - 32 - 8 - 24 DAYS Gel slices Fig. 5. Supersaturation throughout the gel column for different diffusion times. Mother solutions: 1M CaCl, and 1M Na2S04. The base line for /? = 1 detaches the unsaturated and the supersaturated regions. cipitate must always form in the rniddle of the diffusion column, even for different initial con- centrations of mother solutions. However, this prediction stands in contradiction to the experi- mental facts [lo], meaning that either the simplifi- cations introduced are not possible or a comple- mentary hypothesis is necessary. The experimental testing of supersaturation may allow us to check whether the cornmonly made simplifications are excessive. Gel Slices Gel Slices Fig. 6. Supersaturation throughout the gel column for different diffusion times. The base Lines for /? = 1 detach the unsaturated and supersaturated regions. (a) Mother solutions: 1M CaCI, and 0.3M Na2S04; (b) Mother solutions: 0.3M CaC1, and 1M Na,SO,. 68 M. Prieto et al. / Mms-transfer and supersaturarion in crystal growth in gels The supersaturation has been calculated from analytical data by means of the procedure described in section 5. Figs. 5 and 6 show the supersaturation values throughout the gel column for different initial concentrations of mother solu- tions. The results are not adaptable to simple curves. When the initial concentration for both mother solutions is the sarne (fig. 5), the highest super- saturation is reached in slice 15 for all diffusion times. The slight shifting of this maximum from the middle of the column is easily understandable by the lower diffusion coefficient of SO:-, this being the reagent towards which this maximum is displaced. However, when the kncentrations for both mother solutions are different, the maximum supersaturation is not in the middle and its loca- tion changes notable in time. Figs. 6a and 6b show this evolution. Both figures look symmetrical to each other. The 32-day curve maximum is shifted towards the highest concentration reagent, but as time goes by it is progressively displaced towards the reagent of lower concentration. The preceding phenomena may only be explained under the concentration influence in the reagent's diffusion coefficient. This is especially evident in the case of the two mother solutions with very different initial concentrations. More- over, the interaction between the two reagents, diffusing from both ends, produces cross-inter- ference in their respective diffusivities, and this is another factor to take into account. Finally, free ion and complex distribution throughout the gel column may also influence the supersaturation maximum location. At any rate, experimental facts show that the commonly made simpiifications to model the mass-transfer may lead to remarkable deviations from reality. Any interpretation of the first precipitate location must be found in the actual distribution of the supersaturation and other experimental data. For instance, the knowledge of actual concentration distributions is important to check in which region of the gel column the "equality range" condition [5] is fulfilled. Acknowledgement This work was done as a part of project no. 472/84 supported by CAICYT (Ministry of Edu- cation and Science of Spain). Ref erences '* , [l] C. van Leeuwen, J. Crystal Growth 46 (1979) 91. [2] P.T. Cardew, R.J. Davey and J. Garside, J. Crystal Growth 46 (1979) 534. [3] C. van Leeuwen and L.J.M.J. Blomen, J. Crystal Growth 46 (1979) 96. [4] H.K. Henisch and J.M. Garcla-Ruiz, J. Crystal Growth 75 (1986) 195. [5] H.K. Henisch and J.M. Garcla-Ruiz, J. Crystal Growth 75 (1986) 203. [6] H.K Henisch, J. Crystal Growth 76 (1986) 279. (71 L. Arnathieu, Thesis, Universitt d'h-Marseilie 111 (1985). [8] H.E. Lundager Madsen, Nephrologie 5 (1985) 151. [9] A.M. Martell and R.M. Smith, Critical Stability Constants (Plenum, New York, 1974). [lo] J.M. Garcla-Ruiz, Thesis, Universidad Complutense de Madrid (1981).