THE MODEL OF DRYING SESSILE DROP OF COLLOIDAL SOLUTION

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1 THE MODEL OF DRYING SESSILE DROP OF COLLOIDAL SOLUTION I.V.Vodolazskaya, Yu.Yu.Tarasevic Astrakan State University, a Tatiscev St., Astrakan, 41456, Russia tarasevic@aspu.ru e ave proposed and investigated a model o drying colloidal suspension drop placed onto a orizontal substrate in wic te sol to gel pase transition occurs. Te temporal evolution o volume raction o te solute and te gel pase dynamics were obtained rom numerical simulations. Our model takes into account te act tat some pysical quantities are dependent on volume raction o te colloidal particles. Received Day Mont Day Revised Day Mont Day Drops; evaporation; viscosity; pase transition; colloids 1. Introduction Drying sessile drops ave been studied experimentally by a number o groups. Te contact line pinning, te redistribution o te solute particles, ring ormation, sol gel transition, te drop surace deormation were investigated in te works. 1 4 Understanding o te process o drying o solutions is important or scientiic, industrial and medical applications. Several models 1, 5 1 ave been proposed or te deposit ormation and te motion o te liquid-deposit boundary inside a drop. Autors investigated one-dimensional problem utilizing te matter conservation and some geometric assumptions about te drop atmospere interace or te velocity o te pase boundary. In te work 1 autors assumed tat contact line pinning and evaporation are suicient conditions or te deposit ring ormation. en evaporation removes liquid rom te contact line, a low arises to keep te substrate wet up to tat point. Te solute in te drop is dragged to te contact line by tis low, were it accumulates. Te solute particles stop moving wen its local volume raction reaces te tresold value. A zone o immobilized solute (deposit pase) appears at te perimeter and begins to move inward. In te models 5,6,9,1 te deposit pase does not alter te outward liquid lows and evaporation low. In te works 7,8 autors suggested tat ixed solute stops outward lows and evaporation lux rom te surace o te deposit region. In our opinion, models proposed in te works 5,6,9,1 are more adapted to inorganic deposition and models 7, 8 are adapted to colloidal gels.

2 Now te most complicated problem relates to determining te rate o evaporation rom te ree surace o te drop. ell-known unctional orms o te evaporation rate J 1, 11, 1 do not take into account te complex geometry o drop surace and te drop composition. Tus, many researcers 6, 8 use in teir calculations te uniorm evaporation rate J = const or evaporation rate modiications. 7. Model and assumptions Our model is applicable to drops o aqueous protein solution. A drop is deposited onto a orizontal substrate under usual and uniorm environment and is single-pase (sol pase), but some time ater te evaporation begins a second pase appears in te system (gel pase). e assume te volume raction o te solute in te gel pase is ig and its value is ixed. In te sol pase, space averaged volume raction o te solute canges wit time. An axisymmetric tin drop as radius in te substrate plane, R, te eigt o te drop is ( r, t ) ( r is te radial coordinate or distance rom te drop center, t is time), and we assume tat ( r, t) R. During te evaporation process, te contact line does not recede; wic means te quantity R is constant. e assume te drop is rater small, ence surace tension is dominant, and te gravitational eects can be neglected (te ratio ρ gr / σ is smaller tan.5; 5 ere g is te gravitational constant, σ is te surace tension at te liquid air interace). I te colloidal particles are large enoug, diusion can be assumed negligible. e assume tat te luid density ρ is constant and equals te solute density. Hence te sedimentation is negligible. In suc a way te particle velocity equals te luid velocity υ. It is known tat te diusion coeicient o albumin molecules in water is very small: 11 Da = m / s, 13 or te comparison, te ones or te salt NaCl is 9 Ds = 1. 1 m / s. 14 e can utilize te lubrication approximation 15 or te tin drops and slow lows. Fluid low is governed by te Navier Stokes equations: p u = + η, r z (1) p = z wit te boundary conditions () or te pressure p and te radial luid velocity u ( z is te vertical coordinate or distance under te substrate, η is te viscosity): p r σ r, u = =, u = r r z. z= z= z= I set o equations (1) are integrated over te boundary conditions (), te radial velocity and ten te eigt averaged velocity can be written: ()

3 σ u= r z, η r r r r 1 z 1 σ 1 < u>= udz r. = 3 η r r r r Te luid and te solute conservations (4) in te drop wit te boundary and initial conditions (5) give a possibility to obtain te eigt ( r, t ) and eigt averaged volume raction ( r, t ) o te solute: 1 ( r u ) J =, t r r ρ ( ) 1 = t r r t= r u r= r= r= R t= ( r u ) r = 1, R =, =, r= R = u =, =, =. r Here J is te unctional orm o te evaporation rate, is te initial volume raction o te solute (te solute was assumed to be uniormly distributed trougout te drop at t= ), = (,) and initial drop sape is equilibrium. r=. (3) (4) (5).1. Te sol gel transition It is known tat te gelation process o te protein molecules accelerates wen te solute volume raction increases. Te gel pase occurs ater te solute volume raction as reaced te speciic value. Te admixtures in te solution aect te gelation process. Te increase o salt content in te solution screens out te electrostatic interaction between te molecules, and te solution gelation accelerates. 16 Te salt quantity in te solution is described as ionic strengt value. e came to a conclusion tat pase transition sol gel is te gradual concentration transition. Since gel is a solid state, 6 te protein molecules stop moving and gel region keeps its sape. Te luid molecules place among protein molecules or its aggregates and create te connections. Hence it is necessary to include in te model te appearance o te new pase wit dierent pysical properties. Te presence o te gel pase must aect bot te ydrodynamics and te evaporation. In our model te gelation process in sol zone near te liquid air interace is not taken into account. It is known, tat te sol gel transition is accompanied wit te divergence o te viscosity. 4 e suggest or te viscosity η te unctional orm wit te divergence i, or example:

4 b η = = exp a. (6) η Figure 1 sows as a unction o /. Here η is te viscosity o te pure solvent and a and b are natural numbers. It is easy to see tat te value o te parameters a and b are dependent o te protein type and te ionic strengt value ,,4,6,8 1 Figure 1. Te quantity = η / η as a unction o /. a= 1, b= 8... Evaporation rate To maintain a pinned contact line witin te lubrication approximation te evaporative mass lux must be zero at te contact line or liquid drop. In our model we used a ormula tat was suggested in Re. 15: J ( r, t) = J ( A( r R) ) 1 exp 1 / K+ ( r, t) /. (7) e assumed te gel state must decrease te evaporation rate, J, so we inserted into te unctional orm o te evaporation rate (7) te actor wit te unctional orm, or 1 example, (6). 3. Results Te model involves a set o nonlinear partial dierential equations (4) wit boundary and initial conditions (5), takes into account te equations (3, 6, 7) and requires numerical calculation or its analysis. Simulations were run or te dimensionless parameters: te capillary number is η Ca= =.1;.1; 1, ε σρ 3

5 te evaporation number J E = =.1 εη and ε = =. R Te initial volume raction o te solute was =.1;.;.3;.6,.5 =, parameters: a= 1, b= 8, A= 5, K = 1. Our results are presented in te dimensionless orm: te drop eigt was scaled by te and te radial coordinate r was scaled by te R. Te time evolution o te drop eigt proile is sown in Fig. or: t= ;.3 t ;.5 t ;.6 t ;.7 t ;.8 t ; t ; 1.5 t ; t. Here t is te time wen te inal drop proile orms. 1.6,8,6,4,,,4,6,8 1 r Figure. Time evolution o te drop eigt proile (rom top to bottom) =.1, Ca=.1.

6 ,4,3,,1,,4,6,8 1 r Figure 3. Te eigt averaged volume raction ( r, t ) as a unction o r (rom bottom o top). =.1, Ca=.1. Numerical results or te time evolution o te eigt averaged volume raction o te solute ( r, t ) as unction o r are sown in Fig. 3. Te dierent curves correspond to t= ;.3 t ;.5 t ;.6 t ;.8 t ; t. e can see te eigt averaged volume raction o te solute increases wit time. At early times te increase near te contact line is great and te value amounts to.9 at time.5t ere. At later times, te value increases very slowly near te contact line and does not amount to value or te given parameters o te model. Te drop eigt proile decreases and does not vary ater te value reaces to about.8. Tus, we can see te deposit ring proile ormation. Te value at r R in our model as in te Re. 15 tat is te natural consequents o te utilized evaporative lux orm (7). Te eect o te initial volume raction o te solute on te drop eigt proile at time t is sown in Fig. 4. For /. < te inal drop eigt proile as deposit ring near te contact line. Te widt and eigt o te ring increase wit te initial concentration o te solute. In our model, te low is not capable o transerring all o te solute to te contact line, so te liquid zone o te drop contains some solute quantity at all time and tere is gel zone in te central drop part or all investigated values. Te variation o te capillary number Ca ( Ca=.1, Ca= 1) as not a visible eect on te results.

7 ,6,5,4,3,,1,,4,6,8 1 r Figure 4. Te drop eigt proile at time t or te dierent initial volume raction o te solute. Ca= Conclusion In tis paper we investigated processes inside an evaporating sessile two-pase drop. e took into consideration tat te viscosity o a colloidal solution is a unction o volume raction o te colloidal particles dispersed in te solution. Unortunately, to our best knowledge te experimental data regarding surace tension and evaporating lux or te similar systems as not publised yet. As a result, we are not in a position to perorm a direct quantitative comparison wit any experiments Acknowledgments Tis work was supported by te Russian Foundation or Basic Researc, project no r povolzje a. 5. Reerences 1. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. itten, Pys. Rev. E6, 756 ().. R. Deegan, Pys. Rev. E61, 475 (). 3. Yu. Yu. Tarasevic, Pysics-Uspeki. 47, 717 (4). 4. L. Paucard, C. Allain, C. R. Pysique. 4, 31 (3). 5. Yu. O. Popov, Pys. Rev. E71, (5). 6. F. Parisse, C. Allain, J. Pys. II France. 6, 1111 (1996). 7. I. V. Vodolazskaya, Yu. Yu. Tarasevic, and O. P. Isakova, Nonlinear world. 8, 14 (1). 8. T. Okuzono, M. Kobayasi, M. Doi, Pys. Rev. E8, 163 (9). 9. T. A. itten, EPL. 86, 64 (9). 1. R. Zeng, Eur. Pys. J. E9, 5 (9). 11. M. Cacile, O. Benicou, A. M. Cazabat, Langmuir. 18, 7985 (). 1. D. M. Anderson, S. H. Davis, Pysics o Fluids. 7, 48 (1995).

8 13. L. Reyes, J. Bert, J. Fornazero, R. Coen, L. Heinric, Colloids and Suraces B: Biointeraces. 5, 99 (). 14. Tables o Pysical Quantities: A Handbook, ed. by I. K. Kikoin (Atomizdat, Moscow, 1976). 15. B. J. Fiscer, Langmuir. 18, 6 (). 16. L. Paucard, F. Parisse, C. Allain, Pys. Rev. E59, 3737 (1999).