Langmuir 1996,11, 2682-2688
2682
Thickness of the Films Formed from Aqueous Solutions Spread in a Mercury Trough Antony S. Dimitrov, Mariko Yamaki, and Kuniaki Nagayama” Nagayama Protein Array Project, ERATO, JRDC, Tsukuba Research Consortium, 5-9-1Tokodai, Tsukuba, 300-26,J a p a n Received November 15, 1994. In Final Form: April 10, 1995@ Thin wetting films formed by spreading aqueous solutions onto a purified mercury surface yield residual thin layers after drying. The thickness of these residual layers, which had been ellipsometricallymeasured in a previous study, does not significantly depend on the volume of the solution injected onto the mercury that is placed in a trough. Here, we model the successive processes involved in the production of these residual layers: solution spreading, film formation, and film drying. Our model calculations showed that the volume of the wetting film and, hence, the film thickness, does not strongly depend on the volume of the injected solution. The increase in the film thickness that must correspond to an increase in the volume of the injected solution is restrained due to the formation of a Gibbs-Plateau border along the trough wall. A possibly crucial parameter in the formation of two-dimensionalprotein crystals, namely, the time needed for drying a film, is also calculated using our model.
Introduction The formation of two-dimensional (2D) crystalline arrays of different types of lattices from proteins, protein complexes,l viruses,2 and even large organic molecule^^^^ has been recently reported. Two-dimensional crystalline arrays of ferritin,5 ad3 complex of thermophilic ATP synthase,6 chaperonin from Thermus thermophilus,’ and flagellar L-P rings6can be produced by spreading aqueous protein solutions on clean mercury surfaces. Sugars added to the ferritin solution allow the homogeneous distribution of ferritin on the mercury surface and improve the molecular alignment of the hexagonally packed 2D arrays.g A comprehensive review article recently published on the techniques and principles of 2D crystallization of membrane and water-soluble proteins classifies the techniques for 2D crystallization of complex molecules according to the substrates that are used.1° In the article, the main principles in the crystallization on mica, carbon films, mercury, and lipid monolayers are discussed, but the mechanism behind the crystallization is not. In most techniques used in producing 2D crystals, a solution of protein, virus, or organic molecules is spread onto a substrate (which can be either a solid or liquid), and forms a thin wetting film. The film becomes thinner as it dries and, ifthe crystallization is successful,molecular crystals are finally formed. Because the wetting film determines the crystallization conditions, the properties of this film, such as its thickness, are important. Data for the final film thickness on mercury in the absence of protein or other agents to be crystallized were reported
* To whom correspondence may be addressed. Abstract published in Advance ACS Abstracts, June 15, 1995. (1) Harris, J. R. Micron Microsc. Acta 1991,22,341. (2) Horne, R. W. Adv. Virus Res. 1979,24,173. (3) Josefowicz, J. Y.; Maliszewskyj, N. C.; Idziak, S. H. J.; Heiney, P. A.; McCauley, J. P., Jr.; Smith, A. B., I11 Science 1993,260,323. (4) Tomioka,Y.;Ishibashi, M.;Kajiyama, H.;Taniguchi,Y. Langmuir 1993,9, 32. (5) Yoshimura, H.; Matsumoto, M.; Endo, S.; Nagayama, K.Ultramicroscopy 1990,32, 26. (6)Yoshimura,H.; Endo, S.; Matsumoto, M.;Nagayama, K.; Kagawa, Y. J. Biochem. 1989,106,958. (7) Ishii, N.; Taguchi, H.; Yoshida, M.; Yoshimura, H.; Nagayama, K.J. Biochem. 1991,110, 905. (8) Akiba, T.;Yoshimura, H.; Namba, K Science 1991,252, 1544. (9) Yamaki, M.; Matsubara, K.; Nagayama, K.Langmuir 1993,9, 3154. (10) Jap, B. K.; Zulauf, M.; Scheybani, T.;Hefti, A.; Baumeister, W.; Aebi, U.; Engel, A. Ultramicroscopy 1992,46,45. @
by Yamaki et a1.: who found that the final (residual) thickness of the film formed after spreading the solution and evaporating the water does not depend on the sample volume. They stated that a preliminary theoretical analysis showed constant thicknesses of the residual layers for injected solution volumes between 2 and 10 pL, but they did not give a reason for this constancy nor did they explain where the additional amount of the “assemblyaiding agent” is stored. In this report, we model the processes of solution spreading, of the formation of a Gibbs-Plateau border, and of the film drying, to explain why the residual film thickness does not depend on the volume of the spread solution. The model includes evaporation of water from the solution as a function of time during both the solution spreading and film drying and the distribution of the solution volume into the film and Gibbs-Plateau border, which is formed a t the end of the spreading process, when the solution reaches the trough wall. We applied the model to our experiments on the spreading of a ferritin solution and to experimental data from ref 9 on the spreading of sugar solutions. The small dependence of the film thickness on the solution volume injected onto the mercury is due to the formation of a Gibbs-Plateau border, where a large amount of the solution accumulates. The formation of the Gibbs-Plateau border creates a capillary pressure, which restrains the increase in the film thickness by withdrawing solution from the film.
Experimental Section The mercury trough is made of stainless steel. It has axial symmetry and its internal diameter is 24 cm. Two stainless steel barriers restricted the sector area, where the solution was allowed to spread. The trough was filled with dry oxygen gas and was isolated from the atmosphere by a chamber made of stainless steel and glass. The experimental procedures for purifying and activatingthe mercury surface are detailed in ref 5. Spreadingprocedureand film observation. The solution
was injected onto the mercury surfaceusing a Hamilton syringe. The solution spread and covered the mercury surface of area A (between the barriers). Thus, a wetting film was formed. The water evaporated from the solution during its spreading, and continued after the film and Gibbs-Plateau border were formed.
The residual film thickness was measured using an ellipsometer attachedto the mercury trough.5~9In the experiments with ferritin solutions, the dried films were transferred to carbon support
0743-7463/95/2411-2682$09.00/00 1995 American Chemical Society
Langmuir, Vol. 11, No. 7, 1995 2683
Wetting Film Thickness films on electron-microscope grids, which were previously hydrophilized by ion bombardment. After negative stainingwith 2% uranyl acetate for 1 min, the grids were observed at a magnification of 30000x using a JEOL 1200EX electron microscope at a 100-kV accelerating electric field. Materials. Commercially obtained ferritin (horse spleen ferritin,Boehringer, Mannheim, Germany) contained aggregated molecules and subunit fragments. Therefore, the ferritin was purified using a gel-filtration column (S-300, Pharmacia, Sweden). The mercury was of high purity (6-Ngrade,Mitsuwa Pure Chemicals) and was additionally distilled before each experiment. The oxygen was 4-N grade and was not additionally purified. The glucose and the sodium chloride were reagent grade (Wako Chemical, Japan). The samplesofthe ferritin solution that were spread on mercury contained 140 mg/mL (3.18 x mom) ferritin, 0.55 mom glucose, and 0.15 m o m NaC1.
Theoretical Section Water Evaporation. Due to the dry oxygen atmosphere under the trough chamber, water evaporates from the solution during the spreading process and after the wetting film is formed. Here, we propose a simple model to estimate the water evaporation as a function of time. When the solution drop touches the mercury surface, a three-phase contact line is formed. The contact line starts to expand due to the positive spreading coeficientll of the solution on a pure mercury surface. According to eq IV15 in ref 11the distance traveled by the spreading film in time t is proportional to t314. Then, the area on the mercury surface that is covered by the solution is proportional to t3I2 A&) = A"t3/2,
for t < t,; A" = A/ty
(1)
where t, is the time needed by the solution to cover the entire mercury surface, A&) is the covered area a t time t , and A is the total area of the mercury surface. We assumed that after the wetting film has been formed, the area of evaporation is constant and is equal to the sector area between the barriers in the trough, namely, A&) = A . The decrease in the volume Vof the spreading solution on the mercury surface is then,
where,j , is the evaporation flux from a pure water surface (at experimental conditions under the trough chamber), @ is the volume fraction of water, and 12 is a coefficient of proportionality between @ and the surface water fraction. To solve eq 2 for the spreading solution, we substitute A&) by using eq 1 and then integrate to get
v-vs 2 v = Vo - V, In vo- v, - -5 kj&"t5/2
(3)
where V, is the volume of the solution loaded on the mercury surface and V, is the volume of the nonvolatile compounds in the solution. Equation 3 can be transformed into an equivalent form, which is suitable, when V approaches V,
x
Mercury
Figure 1. Schematic of the meniscus profile between the wetting film and trough wall: h is the film thickness, H i s the meniscus capillary rise, 0 is the meniscus-wall contact angle, and q is the running meniscus slope angle.
Equation 4 shows a n exponential decrease in V with t5I2, when V approaches V,. After the wetting film is formed, the water evaporates mainly from the surface of the film with area A . Let now Vfbethe volume of the remaining solution (that forms the residual thin layer after drying) and Vsfbe the volume of the nonvolatile agents in this solution. (We regard in a later section how Vf and Vsf are determined.) Then, by applying the same transformation procedure as for the spreading drop, but for constant evaporation area, A , we obtain the following equations 17 - 17 ' 'sf
V = Vf - V,, In -- kj&(t - t,) Vf - Vsf
(5)
These equations are analogous to eqs 3 and 4, but with a difference that here the time is as t , and in eqs 3 and 4, it is as t512.We used eqs 4 and 6 rather than eqs 3 and 5 due to the divergence of the logarithmic terms when V approaches Vs in eq 3 or Vsf in eq 5. Film Thickness Dependence on the Solution Volume. The solution dropped onto the mercury surface spreads until it reaches the trough wall. Then, a meniscus is formed due to the solution capillary rise along the wall. The meniscus capillary pressure withdraws a part of the solution into the Gibbs-Plateau border, and a thin wetting film is formed on the mercury surface. To simplify our calculations of the thickness of this film, we assumed that (1)the curvature of the cylindrical trough wall is negligible in comparison with the meniscus curvature, (2) the electrolyte in the solution suppresses electrostatic repulsion between the film surfaces, (3)the Hamaker constant ergs12and for water films on mercury is K = -7.22 x (4)the mercury surface is uniformly flat, a s schematically shown in Figure 1. Below, we outline our calculations for the distribution of Vbetween the film and Gibbs-Plateau border. A schematic of the meniscus profile a t the trough wall is presented in Figure 1. The meniscus profile satisfies the Laplace equation of capillarity, which is written below (eq 7) as a set of two e q ~ a t i 0 n s . lAccording ~ to our first assumption, the curvature (sin q/r) is neglected. (The radius r is approximately equal to the radius of the mercury trough.) - d sin 9
dx
+ q2f2,
- PC U
where q =
J?
(7)
dzldx = t a n q
Both eq 3 and eq 4 are valid until the spreading solution reaches the trough walls. Equation 3 shows that the decrease in V depends on time as t5I2when V is close to V,, namely, in the beginning of the spreading process.
Here, Q, is the running slope angle, P, is the capillary pressure a t the film-meniscus contact line, q is the
(11)Adamson, A. W. Physical ChemistyofSurfaces,5thed.;A Wiley-
(12) Usui, S.; Sasaki, H.; Hasegawa, F. Colloids Surf 1986,18,53. (13) Princen, H.In Surface and Colloid Science; MatijeviC, E., Ed.; Wiley-Interscience: New York, 1983; Vol. 2, p 1.
Interscience Publication: New York, 1990; Chapter IV.
Dimitrou et ai.
2684 Langmuir, Vol. 11, No. 7, 1995 capillary constant for the solution-gas interface, AQ is the solution-gas density difference, g is the gravity acceleration, and Q is the surface tension of the solution. At a known contact angle, 8,between the meniscus and the stainless steel trough wall, we calculate the value of the capillary rise, H , by integrating eq 7
Equation 8 is used later to calculateH, which is a boundary condition in eq 11. The Gibbs-Plateau border volume, V, is obtained by integrating the following equation avmidx = L+)
(9)
where L is the perimeter of the mercury trough. We eliminate x by combining eqs 7 and 9, and thus obtain the followingset oftwo equations, one for the meniscus profile and one for V,,,
avmia9= - Lz(p,)cos p,
5 + q22
(10)
0
These equations can be numerically integrated using the following boundary conditions.
Therefore, we obtain V, when p is equal ton, namely, V, = Vm(n).However, the total solution volume in the trough Vis the sum ofthe two volumes, namely, that in the GibbsPlateau border and that in the film V=V,+Ah
(12)
where h is the film thickness. When the film-meniscus mechanical equilibrium is established, the capillary pressure, Pc, is equal to the film disjoining pressure, ll. According to assumption 2 above concerning the suppression of the electrostatic component of I'I, the disjoining pressure is supposed to be determined by the van der Waals forces. Therefore, we use the following expression for the capillary pressure, with assumption 3 concerning the Hamaker constant of water films on mercury.12
p , = n = - --K - 7.22 x 6Zh3
6nh3
(13)
Equation 13, together with eqs 10 and 12, completes the set of equations determining the distribution of the solution volume between the film and Gibbs-Plateau border, and the dependence of the film thickness on the solution volume. Then, a n analytical solution of eq 10 for the volume of the Gibbs-Plateau border, Vm,reads Vm = 7 L sin qo
Q
+q
Lpc
{ ( a + 2)[E(S, r ) -
3 a m
E(?, r ) ] - a[d$,r ) - F ( 2 , r ) ] } (14) where the functionsF and E are the first and second elliptic
integrals, re~pective1y.l~ The integral module, r, and the parameter, a , are explicitly given a s follows:
The elliptic integrals must be solved numerically. Under certain conditions it is easier to integrate the set of equations seen in eq 10 than to calculate the integral functions E and F. In our further calculations, we solve both eqs 10 and 14 to ensure the correctness of the numerical procedures. The Residual Film Thicknesses. We calculated the residual thicknesses of the films formed in the mercury trough according to the following scheme. The decrease in the injected solution volume is calculated at time t = t, by using eq 4 (which does not diverge as eq 3 does, when V approaches V8h To solve eq 4, we first calculate the volume of nonvolatile substances, V,, from the solution concentrations, then we determine the spreading time, t,, from the experiments, determine the constant A" from the third equality in eq 1,and determine the product kj, by varying its value to adjust the evaporation time, namely, the time for the film on the mercury surface to dry. Mathematically, the nonlinear eq 4 is solved by applying numerical methods. After spreading, when the solution reaches the trough wall, the Gibbs-Plateau border is formed and the volume of the remaining solution, V, calculated from eq 4 a t t = t, is divided into two parts, namely, volume of the film, V, and volume of the GibbsPlateau border, V,. The volume V, is calculated according to eq 14. Independent determination of V, is made by numerical integration of eq 10, with the boundary conditions given in eq 11. Then, we suppose that the tangential mobility of the mercury surface decreases due to rapid adsorption of the sugars or protein. Considering the hydrodynamic viscous friction in thin films, we assume that there is no flow of the solution from the Gibbs-Plateau border back into the film. Therefore, evaporation of water during the film drying, which must be accounted for in the model, is restricted to the film area A. We calculate the decrease in the film volume, V, and, respectively, in the film thickness, h, according to eq 6 (which is solved similarly to eq 4). In this equation, Vfis the volume of the solution remaining in the film after the solution has been separated into Vm and Vf. Then Vsfis calculated by using the relation
(16) which, in fact, represents the distribution of the nonvolatile substances as a part of the solution. The residual film thickness that is ellipsometrically measured can also be calculated from the above relation a s h = V,dA. However, by solving eq 6 we also calculate the evaporation time, which we compare later to experimental evaporation times reported in ref 9. We prepared and used a computer program that follows the above procedure. For the smaller volumes of each solution spread on the mercury, we input different trial values for the evaporation rate product kje until the calculated evaporation time was adjusted to the one obtained experimentally. Then, using the same value of kje,we calculated the evaporation times for the larger volumes of spread solution and compared them to those obtained experimentally. (14) Janke, E.; Emde, F.; Liisch, F. Special Functions, 3th ed.; Nauka: Moscow, 1977;p 93;Tafeln Hoherer~unktionen;B.G. Teubner Verlagsgesellschaft: Stuttgart, 1960.
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Langmuir, Vol. 11, No. 7, 1995 2685
v r ......................... 0
1
2
3
4
Solution volume, V (pl)
;. ...,....,
5
8
7
Time, t ( s )
h
f
.c vr”
35[
301
c t
U
s
t
, ,!
16t
Surface tension, u (dynkm)
’. -c
--*--+---*---*--4
l5
L 40 45 50 55 60 65 70
2035
6.0
5.5
8.0
0.6
7.0
Time, t (8) Figure 2. An example of the calculated (A) volume of the solution that spreads, V, and (B)thickness of the film that dries, h, as a function of time, t , when a drop with volume of 2 p L (0)or 10 p L (A)is loaded on the mercury surface.
Results We applied and detailed the procedure for calculating residual film thicknesses to the experiments that involved the spreading of 2-pL and 10-pL ferritin solution drops into the mercury trough. Then, we calculated the residual film thicknesses for the experimental data reported by Yamaki et aL9 and compared them to the experimentally measured values. SolutionVolume Decrease during the Spreading Process. While the ferritin solution injected onto the mercury surface spreads, the water evaporates and the solution volume decreases according to eq 4. According to eq 1 for A = 430 cm2 and t, = 5 s we calculated A = 38.5 ~ m ~ l s From ~ ’ ~ .the solution concentrations we calculated V, = 0.41 p L when Vo is 2 pL and V, = 2.05 pL when V, is 1OpL. (Note that in all calculations we assumed all densities are equal to 1, instead of 1.55 for glucose and 1.26 for glycerin, for simplicity and because the density assumption has only a small effect when compared with the other crude assumptions, e.g., separation of the spreading process into three successive, distinct steps.) The product, kje, in eqs 4 and 6 was determined from the experiments on the spreading of 2-pL solution drops, where we observed that the wetting film dried just as it reached the trough wall, and, hence, the water evaporated within 5 s. By varying kje so that the water evaporation cds. terminated within 5 s, we obtained kje = 5.5 x This value was then used to calculate the dependence of the solution volume on time, when 10-pL solution drops were injected into the trough. For this same evaporation time (Le., t, = 5 s), the volume, V, decreased from 10 to 7.0 pL, when the solution reached the trough wall and a
3
40
4
30
-E
20 10 30
60
90
Contact angle, 0 (deg)
Figure 3. Calculated thickness of the film that is formed at the spreading end as a function of the (A) volume of the solution remaining on the mercury surface, (B)surface tension of the solution,and (C) contact angle between the solution and trough wall.
wetting film was formed. The time dependence for the calculated spreading solution volume is plotted in Figure 2A for the 2-pL solution drops (by open circles) and for the 10-pL drops (open triangles) injected into the trough. Gibbs-Plateau Border and the Thickness of the Wetting Film.When the solution reaches the trough wall, it is rapidly distributed into two parts: the wetting film and Gibbs-Plateau border (see, e.g., Figure 1). The area of the wetting film in these experiments was A = 430 cm2 and the length of the trough wall was L = 96 cm. The contact angle, 8, between the solution and trough wall we assumed to be 45O, when the meniscus rose up along the wall at height, H. This value of the contact angle was that measured for a sessile drop formed from the same solution on a stainless steel surface. We assumed the solution surface tension, (T,to be around 50 dydcm, the same as for a solution outside the trough. The results obtained for the film thickness calculated according to eqs 12 and 14 are presented in Figure 3. Figure 3Ashows the dependence of the film thickness, h, on the solution volume, V, remaining in the trough at the end of the spreadingprocess (open circles and the left ordinate axis). Also in Figure 3A is the dependence of the film volume, Vf=Ah, on V(open triangles and the right ordinate axis). Our numerical results show that when Vis less than 1pL, most of the solution remains in the film, and for V from 4 to 10 pL, most of the solution is withdrawn into the
Dimitrov et al.
2686 Langmuir, Vol. 11, No. 7, 1995 Gibbs-Plateau border and the film thickness, h, is almost constant, between 25 and 30 nm. During the spreading process (T is slightly higher than its equilibrium value due to the hydrodynamic mass exchange between the bulk solution and the surface. Due to relaxation processes, the value of (T decreases after the film formation. The variations of a were accounted for in the film thickness calculations, although, great variations were not expected due to the high initial concentrations of the injected solutions. Figure 3B shows that h does not strongly depend on a; the total change of h is only 5 nm, from 30 to 25 nm, when avaries within a 25 dyn/cm range, from 40 to 65 dyn/cm. Due to the hysteresis of the contact angles on solid substrates (for details on the hysteresis see, e.g., the book by Adamson,ll Chapter VII) some variations in the value of the contact angle 6 were expected and their influence on the film thickness was accounted. Figure 3C shows the dependence ofthe film thickness on the contact angle, 6, for 6 varying between 15" and 75", and confirms that the calculated film thickness, h, does not strongly depend on 6 for contact angles smaller than 50". The variations in the film thickness, h, as functions of u and 6 were calculated for a fixed value of the solution volume, V= 7.0 pL, which was estimated to remain on the mercury surface a t the end of the spreading process, when 10-pL ferritin solutions were injected into the mercury trough. The main results demonstrated in Figure 3 are (1)when the spreading solution contacts the trough wall, a part of the solution is withdrawn into the Gibbs-Plateau border, and (2) the thickness of the formed film does not significantly depend on the solution volume, solution surface tension, and wettability of the trough wall. For the spreading process of the 10-pL solution, our model calculations showed that 5.8 pL from the remaining 7.0 pL is distributed to the Gibbs-Plateau border and 1.2 pL remains in the wetting film. At the end of the spreading process and after the formation of the wetting film and Gibbs-Plateau border, we calculated the film thicknesses to be as follows: 9.5 nm when 2-pL drops were spread (0.42 pL remained to be distributed after the spreading) and 28.6 nm when 10-pL drops were spread. Film thickness decrease during the film drying is plotted in Figure 2B. In fact, this is a n expansion of the right side of Figure 2A, where the film thickness was obtained as h = V/A and V was calculated as function of time according to eq 6. For the plot in Figure 2B the film volume, Vf, was equal to 1.2 pL, namely, the amount remaining in the film after separating the volume of the spreading solution into two parts. The volume of the nonvolatile solution compounds remaining in the film, Vsf,was calculated according to eq 16. The residual film thickness, h = 9.4 nm (t > 7 s), calculated for the 2-pL drops is larger than that calculated for the 10-pL drops, h = 8.4 nm (t > 7 SI. The values of h calculated as a function of V,, are plotted in Figure 4. Our calculations for the above solution showed that the residual film thickness must have a local, relatively sharp maximum, when 3-pL solutions are injected and flat minimum, when 16-pL solutions are injected. Furthermore (see the left inset in Figure 41, residual film thickness slightly increases as the spread solution volume increases up to 3.5pL. Above this value, the buffering ability of the Gibbs-Plateau border is saturated and h rapidly increases as V,, increases (see the right inset in Figure 4). Ferritin Concentration on the Mercury Surface. The calculation results for residual ferritin film thicknesses on mercury agrees with the experimental data shown in Figure 5. The surface concentration of ferritin
e
101
0
5
0
15
10
20
Injected volume, V,, ( pl ) Figure 4. Residual film thickness calculated as a function of the solution volume injected onto the mercury surface. Extension of the curve for larger solution volumes is shown in the insets. The minimum thickness was calculated for V, rz 16 pL (excluding V, 0). Table 1. Distribution of Assembly-AidingAgents on a Mercury Surface (From Table I in Reference 9) expt no.
11
thickness aft r loaded coverage evaporation volume area evaporation &L) (%I time average range
(1)
agent
0.55M
2
80
308
2.7 f 1.5 0-100
10
100
120 s
3.5 f 1.6 3-199
2
100
10min
8.9 f 0.6 8-10
10
100
20 min
7.8 iz 0.4 7-8
glucose
12 0.55M glucose 17 0.55M glycerol 18 0.55 M glycerol
in Figure 5A is higher than that in Figure 5B. To clarify our statement, we also calculated the surface ferritin concentration rather than the model film thickness, thus connecting the model calculations to the obtained experimental data. From the initial ferritin concentration in the solution, we calculated each microliter to contain 1.9 x 1014ferritinmolecules. When a 2-pL solution was spread on the mercury, the meniscus was not formed and the total ferritin amount was distributed on the mercury surface with a n area of 430 cm2. Thus, the ferritin surface concentration was calculated to be 8.9 x 10'' molecules/ cm2,which corresponds to almost complete surface coverage, 1.0 x 10l2 molecules/cm2 according to Nygren and Stenberg.15J6 When a 10-pL solution was spread, the meniscus was formed and the arisen capillary pressure caused the solution to distribute to the Gibbs-Plateau border as well. Using our model and eqs 4, 12, and 14, we calculated the volume of the solution that remained in the film to be 1.2 pL, which is 5.8 times smaller than the total solution volume (V = 7.0 pL, according eq 4) a t the end of the spreading process. Therefore, 17% of the total ferritin amount (1.9 x 1015molecules)was distributed to the mercury surface and 83% to the Gibbs-Plateau border. Then, we calculated that the surface concentration of ferritin, 7.6 x 10'' molecules/cm2,was less than that obtained after the spreading of 2-pL solution drops, which agrees with the experimental data in Figure 5. Residual Film Thicknesses Measured after the Spreading Sugar Solutions. Here we apply our model to previous experimental data reprinted in Table 1from ref 9. In these calculations, we assumed the spreading time, t,, to be the same as that in our experiments. We (15)Nygren, H.; Stenberg, M. Biophys. Chem. 1990,38, 67. (16)Nygren, H.; Stenberg, M. Biophys. Chem. 1990,38, 77.
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Langmuir, Vol. 11,No. 7,1995 2687
Figure 5. Transmission electron micrographs of 2D arrays from horse spleen ferritin molecules. (A) Dense-packedarrays obtained when 2-pL solution drops were spread on the mercury surface. A single domain is seen in the whole area. (B) Arrays obtained when 10-pL solution drops were spread on the mercury surface. The scale bars are 50 nm.
then calculated the products connected with the rate of c d s for the glucose evaporation, kje,to be 1.8 x solution (for the measured evaporation time of 30 s) and 4.0 x c d s for the glycerol solution (for the measured evaporation time of 600 s). We calculated h to @e25 A when 2 pL of the glucose solution was spread, 31A when 10pL of the glucose solution was spread, 11A when 2 pL of the glycerol solution was spread, and 15 when 10 pL of the glycerol solution was spread. For the spread 10-pL solutions, we calculated evaporation times of 40 s for the glucose solution and 900 s for the glycerol solution (according to eq 6, when the relative difference between V and V,f fell below Our model calculated h values that agree with experimentally measured values for the spreading of glycerol solutions and values that are about 10 times higher for the spreading of glucose solutions. Similarly, our model calculated evaporation times that agree with experimentally measured values for the glycerol solutions, and about 4 times smaller for the glucose solutions. Despite these discrepancies, our model showed a slight film thickness dependence on the volume of spread glucose solutions.
Discussion The idealized spreading process created here in our model explains why the residual film thickness is almost independent of the volume of the injected solution, although all the details of the actual spreading process were not taken into account. For example, when the spreading solution reaches the trough wall, we assumed that an equilibrium distribution of the solution between the film and Gibbs-Plateau border is immediately reached. In fact, due to the large area of the trough, the time for the distribution of the solution to take place should be accounted for, and a more detailed hydrodynamic approach must be taken. However,considering the initial fluidity of both film surfaces (air-water and mercurywater), high capillary pressure across the formed meniscus, and good agreement between the calculated and experimental data, we believe that the assumption of immediate distribution of the solution to the film and to the Gibbs-Plateau border sufficientlymodels the complete spreading process. We assumed that the solution does not flow back to the film during the film drying and after forming the Gibbs-
Plateau border, or a t least not back to the spots where ellipsometric measurements are done, and which were placed quite far (more than 3 cm) from the trough wall. The solution does not flow back to the film due to the rapid adsorption (in 2-3 s) of sugars onto the mercurywater interface, where the adsorbedsugar layer can lessen the tangential mobility of the surface and even “freeze” the surface after the wetting film is formed. The hydrodynamic friction in such wetting films (on tangentially immobile substrates) is significant, keeping in mind the viscosity of the sugar solutions. Then, taking into account the relatively short measured time of water evaporation (on the order oft,), the assumption that the solution does not flow from the Gibbs-Plateau border back to the film seems reasonable. One possibility for the discrepancies between the calculated and experimental parameters for the glucose solutions injected into the mercury trough is the formation of relatively stable crystalline hydrate that does not release water below 50 “C. Another possibility is the assumed value for the Hamaker constant of water, 4.38 x erg: other theoretical calculations give different values, erg. Also, this constant erg to 4.35 x from 0.6 x increases during the water evaporationdue to the increase in the concentration of the solution. The existence of electrostatic disjoining pressure, which was neglected in our calculations, can also be a reason for 10-folddifference between the calculated and measured film thicknesses. Most probably, however, this discrepancy is due to the nature of the spreading glucose solutions, namely, the spreading of glucose solutions is not even over the substrate surface and the thickness of the formed film varies over a wide range (Table 1and ref 9).
Conclusions In this report, we studied and modeled the spreading process of aqueous solutions in a mercury trough to obtain the residual film thickness on the mercury surface. For the modeling, we separated the entire spreading process, from the drop injection to the film drying, into three successive parts: spreading, film forming, and film drying. Our model consists of three components: water evaporation from the spreading solution, film thickness dependence on the solution volume in the trough, and water evaporation during the film drying.
Dimitrou et al.
2688 Langmuir, Vol. 11, No. 7, 1995
The model spreading process was applied to three different solutions: glycerol, glucose, and ferritin. Our model shows that when a large amount of a solution (e.g., 10-pL drops) is spread onto the trough, a Gibbs-Plateau border is formed along the trough wall, where the greater part of the solution accumulates, thus inhibiting the increase in the residual film thicknesses when the volume of injected solution is increased. The model explains the residual film thicknesses of glycerol on mercury and predicts the surface concentration of ferritin molecules on the mercury surface (Figure 5). Our results provide insights into the influence of film properties on the crystallizationprocess. Future develop-
ments include clarifying the crystallization process on a substrate and how, by only increasing the volume of the injected ferritin solution, a square-type lattice starts to form (Figure 5B)in addition to the hexagonal one (Figure 5A).
Acknowledgment. We thank Dr.Peter A. Kralchevsky from the Laboratory of Thermodynamics and Physicochemical Hydrodynamics, Faculty of Chemistry, University of Sofia, who helped simplify the real geometry of the mercury trough. LA9409081
