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Diffusion studies in silica gels
Diffusion Studies in Silica Gels JAMES C. D E N N I S Physics Department, Stephen F. Austin State College, Nacogdoches, Texas
Silica gel structure has interested scientists In a previous paper (1), a method was for nearly a hundred years. Most of the proposed whereby diffusion coefficients of interest has centered around the control, electrolytes can be obtained. The possible applications, and characterization of the results of such an investigation have now enormous surface-to-volume ratios that gels been extended to include an order of magnipossess. A typical ratio is 500 m2/gm for dry tude estimate of the pore size in a gel. Diffusilica gel. Since surface effects are responsible sion coefficients are measured by analysis of for silica gel's desirable properties, numerous constantly stirred electrolyte placed on the determinations of surface areas, pore sizes, surface of a gel. If the pore sizes are so large and pore densities have been made. On the that the solid portion of the gel does not other hand, quantitative data on hydrogel interfere with the normal diffusion process structure are not plentful in the literature. (the blocking effect is negligible), there is a Two reasons for this de-emphasis on hy- simple relationship between the aqueous difdrogels are (1) the hydrogel is not as useful, fusion coefficient and the gel coefficient, on and (2) there are no standard techniques that the assumption that only volume effects are can be applied to the problem. Gas adsorp- present: tion, X-ray, and density methods that can be Dg~l = XD, [1] applied to the problem of dry gel structure are not suitable for hydrogel study. where X is the ratio of pore volume to total With regard to (1), interest in silica volume and D is the aqueous diffusion at the hydrogel has recently experienced a slight temperature measured. Thus it would be a upsurge because of its potential as an ideal relatively simple matter to find the total medium for low temperature crystal growth. solid volume of the gel, an impo%ant bit of Some striking examples of crystal growth in information in itself, if only volume effects gels have been reported, and lately optically were present. In gels of interest, though, surperfect 1 era. hexagons of lead iodide less face diffusion enhanced pore transport evithan 20 microns thick have been grown in dently occurs, and the diffusion coefficient silica gels in our lab. Exactly how the gel becomes promotes sparse nucleation and regular, imD~el -~ XflD, [2] purity-free growth is not clearly understood. It will be seen that a reduction in diffusion where 3 is a surface diffusion factor > 1 rates, which might account for better growth which depends slightly on diffusant concenof crystals, is not in evidence. Obviously, tration and silica concentration, and hence knowledge of silica hydrogel structure would must be suppressed if h is to be taken from aid in analysis of the gel effect, but (2) is now the data. a stumbling block. Consequently, it is proposed that the method for study be a CONCENTRATION diffusion one, which, incidentally, involves a When speaking of concentration in a gel, minimum of equipment. It should be understood here that emphasis is more on a tenta- it is helpful to specify exactly what is meant. tive approach than on the presentation of At least three distinct concentrations apply to gels. They are (1) the pore concentration, absolutely reliable results. 32 Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
DIFFUSION STUDIES IN SILICA GELS which varies from pore to pore, and which may depend on the variation in pore sizes within a sample; (2) the "true" concentration, the average pore concentration at any level; and (3) the " t o t a l " concentration, which denotes the mass of diffusant per refit total volume, including any adsorbed material. The gel diffusion coefficient is a measure of the "total" concentration gradient. In a like manner, the flow rates are defined. At the gel-solution interface, (2) is continuous, and hence this boundary condition determines the form of the theoretical diffusion equation. PORE VOLUME (X) It should be pointed out that the pore volume to be determined excludes adsorbed material initially present in the gel, i.e. in most cases a layer of hydrated sodium ions. The ratio is related to the densities of the solid portion of the gel and the liquid in the pores by pV = mV, + p~V~, or
p V = psVs + pr(V -
V~),
since V=V~+V~,
33
where p~ is the density of the solid, p is the density measured by displacement, and pr is the density of the solution in the pores. The preceding expression becomes X
=
Ps
--
pips
--
[3]
Pr.
Thus knowledge of X, p, and p,. allows one to determine m , the silica density. T H E O R Y AND R E S U L T S
Table I is a tabulation of Eqn. [4], which expresses the change in concentration of a constantly stirred diffusing electrolyte as a function of time. The theoretical expression is derived for constant temperature and equal volumes of diffusant and gel. It is v(t)
-
2
,~=o (2r~ + 1 ) ~ ~ [4]
.exp(
(2n +' l ) 2 47 r 2 R )
where V(t) is the fraction of initial concentration that has passed across the interface at the time t measured, and R = Dge¢/a 2 (a is the length of the gel or solution column). Any number of concentration fractions can be taken as a function of time; corresponding R values can be taken from Table I, and
TABLE I V(R) v s . R R
V(R)
R
v(R)
R
V(R)
R
V(R)
0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038
0.0353 0.0357 0.0377 0.0505 0.0534 0.0618 0.0668 0.0714 0.0757 0.0798 0.0837 0.0874 0.0910 0.0944 0.0977 0.1009 0.1040 0.1070 0.1100
0.040 0.042 0.044 0.046 0.048 0.050 0.052 0.054 0.056 0.058 0,060 0.062 0.064 0.066 0.068 0.070 0.072 0.074 0.076
0.1128 0.1156 0.1183 0.1212 0.1236 0.1262 0.1287 0.1311 0.1335 0.1359 0.1382 0.1405 0.1423 0.1449 0.1471 0.1493 0.1514 0.1535 0.1555
0.078 0.080 0.10 0.12 0.14 0.16 0.18 0.20
0.1576 0.1596 0.1784 0.1954 0.2111 0.2256 0.2342 0.2520 0.2650 0.2756 0.2865 0.2968 0.3066 0.3159 0.3248 0.3333 0.3143 0.3489 0.3562
0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80
0.3631 0.3679 0.3760 0.3820 0.3877 0.3931 0.3981 0.4031 0.4078 0.4122 0.4165 0.4205 0.4242 0.4279 0.4314 0.4347 0.4409 0.4437 0.4470
0.22
0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42
Journal of Colloid and Interface Science, Vol. 28, No. 1, September t968
34
DENNIS TABLE II
GEL DIFFUSION
COEFFICIENTS
(×105
CM 2 GM) OF
~ I V E E L E C T R O L Y T E S IN T H E A , B , AND C GELS Gel
A B C Water
(25°C)
NaC1
KBr
KCI
CaCh
KI
2.2 2.0 2.2 1.5
2.9 3.1 2.9 2.0
2.8 2.7 2.7 1.9
1.9 2.7 1.9 1.2
2.9 2.8 3.1 2.1
subsequent D's can be averaged for a best value. Table II gives the gel diffusion coefficients (average of ten values, average deviation 4 - 5 % ) of five electrolytes in hydrochloric, tartaric, and acetic acid gels, referred to, respectively, as gels A, B, and C. The standard gel, hereafter referred to as the " S " gel, contained 0.125 gm/cc of Si02 from Na~Si03. 5H20. A constant hydrosol p H of 5.5 was maintained by mixing the neeessary amounts of acid with the silicate. It is seen that in the " S " gels diffusion coefficients, and hence diffusion rates, are greater than those pertaining to aqueous solution. This result rules out the possibility of reduced flow rates causing good growth of crystals in gels. I t seems likely that some other phenomenon such as the absence of convection currents is responsible. If Xfl is nearly constant for a particular gel, the equivalent statement, D g e l / D w ~ = constant, should hold. I t is easily seen that the ratio does not v a r y much for a particular gel. The subject of interest is not the ratio, but X, and so gels with fl = 1 are needed. When the silica concentration is increased to 1.5S, the ratio decreases to less than one. If it is assumed that surface diffusion becomes negligible (fl = 1) in such gels, then X can be obtained from gels of high silica content ( > 1.5S), and an extrapolation can be made to S gels. Figure 1 shows an experimentM result, an eneouraging extrapolation to almost 100 % pore volume for zero silica eoneentration. Each point on the graph is an average of pore space determined by each electrolyte in Table I. Agreement was good among the five values except for CaC12, whieh consistently gave X's 30 % higher than the other salts. Possibly the extrapolation is
not valid when the diffusing ions are doubly charged. I t is seen that the pore volume (which excludes the volume occupied by the adsorbed ions) comprises about 80 % of the S gel total volume. PORE SIZE DETERMINATION At this stage of the investigation, it must be re-emphasized that assumptions upon which the following technique is based are probably incomplete. Future invsetigations should entail a more thorough analysis. Assumptions necessary are (a) a regular, ideal, gel model, and (b) the presence initially of an adsorbed monolayer of hydrated sodium ions, which can be removed b y base exchange. The ideal gel will consist of long, uniform tubes parallel to each Cartesian direction. Each "pore" is the intersection of three channels, and will be considered as a sphere of radius r, the width of each channel. The distance between pores is d. If r >> d, the structure consists of small gel particles connected by thin threads. Consider a cube of gel i cm on a side, with n tubes along each direction. There are n 2 tubes on a side, and 3 n ~ in the unit cube. The total volume of pore and channel space is X = 3n2~rr 2 - -
2n3(4~Irr3).
[51
The second term on the right is the overlapping correction to the volume. Since there
•" G E L A LO
•
-.
0.~
o
B
I:I
C
-,%,,
0.7
~. ~ z~
w0.6
>04 o 0.3 02 0.1 0,5 SILICA
1.0 I;,5 CONCENTRATION
2D 2.5 IN S UNITS
5.0
FIG. 1. Average pore volume vs. silica concentration for the A, B, and C gels.
Journal of Colloid and Interface Science, Vol. 28, No. I, September 1968
35
DIFFUSION STUDIES IN SILICA GELS TABLE III PORE VOLL~iVfES~SIZES~ AND DENSITIES Gel
Xl
X2
nr
n(r - 5.1)
r(A)
d~)
n(X 10~)
A B C
0.82 0.77 0.80
0.97 0.97 0.97
0.385 0.385 0.385
0.360 0.345 0.354
79 49 65
47 30 39
4.9 7.7 5.9
are three rows, the volume of each intersection is accounted for three times, and therefore must be subtracted twice. Equation [5] yields the quantity (nr) when a numerical X is taken from Fig. 1. In order to determine separately n and r, their product must be determined from a gel with the monolayer removed. For this purpose, gels are base exchanged with 1% NH4C1 for one week, and then treated daily with fresh distilled water for one week, in order to remove the exchanged cations. In the treated gels the result is the determination of the quantity n(r - 5.1), 5.1 being the hydrated sodium ion diameter in Angstrom units. Finally, the pore radius and the linear density are calculated from the ratio n r / n ( r - 5.1). Table I I I lists the experimental results. The pore volumes before and after exchange, respectively, are X~and X2. Evidently the slight differences in effective pore space
are due to differences in the thic!mess of the adsorbed layer. The pore size of about 50 A may be compared with Plank's average value of 50 A for the dried gel (2). T E M P E R A T U R E D E P E N D E N C E OF T H E GEL D I F F U S I O N C O E F F I C I E N T
Initial work has revealed that the variation of the diffusion coefficient with temperature is not monotonic. As the temperature increases, surface diffusion decreases, but with further increase, the normal increase with temperature of the diffusion coefficient takes place. Thus the diffusion coefficient has a maximum value at some temperature. REFERENCES 1. DENNIS, J., To appear ill J . C h e m . E d u c . 45, 432-433 (1968). 2. PLANK, C. J., J . C o l l o i d S c i . 2,399 (1947).
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
Silica gel structure has interested scientists In a previous paper (1), a method was for nearly a hundred years. Most of the proposed whereby diffusion coefficients of interest has centered around the control, electrolytes can be obtained. The possible applications, and characterization of the results of such an investigation have now enormous surface-to-volume ratios that gels been extended to include an order of magnipossess. A typical ratio is 500 m2/gm for dry tude estimate of the pore size in a gel. Diffusilica gel. Since surface effects are responsible sion coefficients are measured by analysis of for silica gel's desirable properties, numerous constantly stirred electrolyte placed on the determinations of surface areas, pore sizes, surface of a gel. If the pore sizes are so large and pore densities have been made. On the that the solid portion of the gel does not other hand, quantitative data on hydrogel interfere with the normal diffusion process structure are not plentful in the literature. (the blocking effect is negligible), there is a Two reasons for this de-emphasis on hy- simple relationship between the aqueous difdrogels are (1) the hydrogel is not as useful, fusion coefficient and the gel coefficient, on and (2) there are no standard techniques that the assumption that only volume effects are can be applied to the problem. Gas adsorp- present: tion, X-ray, and density methods that can be Dg~l = XD, [1] applied to the problem of dry gel structure are not suitable for hydrogel study. where X is the ratio of pore volume to total With regard to (1), interest in silica volume and D is the aqueous diffusion at the hydrogel has recently experienced a slight temperature measured. Thus it would be a upsurge because of its potential as an ideal relatively simple matter to find the total medium for low temperature crystal growth. solid volume of the gel, an impo%ant bit of Some striking examples of crystal growth in information in itself, if only volume effects gels have been reported, and lately optically were present. In gels of interest, though, surperfect 1 era. hexagons of lead iodide less face diffusion enhanced pore transport evithan 20 microns thick have been grown in dently occurs, and the diffusion coefficient silica gels in our lab. Exactly how the gel becomes promotes sparse nucleation and regular, imD~el -~ XflD, [2] purity-free growth is not clearly understood. It will be seen that a reduction in diffusion where 3 is a surface diffusion factor > 1 rates, which might account for better growth which depends slightly on diffusant concenof crystals, is not in evidence. Obviously, tration and silica concentration, and hence knowledge of silica hydrogel structure would must be suppressed if h is to be taken from aid in analysis of the gel effect, but (2) is now the data. a stumbling block. Consequently, it is proposed that the method for study be a CONCENTRATION diffusion one, which, incidentally, involves a When speaking of concentration in a gel, minimum of equipment. It should be understood here that emphasis is more on a tenta- it is helpful to specify exactly what is meant. tive approach than on the presentation of At least three distinct concentrations apply to gels. They are (1) the pore concentration, absolutely reliable results. 32 Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968
DIFFUSION STUDIES IN SILICA GELS which varies from pore to pore, and which may depend on the variation in pore sizes within a sample; (2) the "true" concentration, the average pore concentration at any level; and (3) the " t o t a l " concentration, which denotes the mass of diffusant per refit total volume, including any adsorbed material. The gel diffusion coefficient is a measure of the "total" concentration gradient. In a like manner, the flow rates are defined. At the gel-solution interface, (2) is continuous, and hence this boundary condition determines the form of the theoretical diffusion equation. PORE VOLUME (X) It should be pointed out that the pore volume to be determined excludes adsorbed material initially present in the gel, i.e. in most cases a layer of hydrated sodium ions. The ratio is related to the densities of the solid portion of the gel and the liquid in the pores by pV = mV, + p~V~, or
p V = psVs + pr(V -
V~),
since V=V~+V~,
33
where p~ is the density of the solid, p is the density measured by displacement, and pr is the density of the solution in the pores. The preceding expression becomes X
=
Ps
--
pips
--
[3]
Pr.
Thus knowledge of X, p, and p,. allows one to determine m , the silica density. T H E O R Y AND R E S U L T S
Table I is a tabulation of Eqn. [4], which expresses the change in concentration of a constantly stirred diffusing electrolyte as a function of time. The theoretical expression is derived for constant temperature and equal volumes of diffusant and gel. It is v(t)
-
2
,~=o (2r~ + 1 ) ~ ~ [4]
.exp(
(2n +' l ) 2 47 r 2 R )
where V(t) is the fraction of initial concentration that has passed across the interface at the time t measured, and R = Dge¢/a 2 (a is the length of the gel or solution column). Any number of concentration fractions can be taken as a function of time; corresponding R values can be taken from Table I, and
TABLE I V(R) v s . R R
V(R)
R
v(R)
R
V(R)
R
V(R)
0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038
0.0353 0.0357 0.0377 0.0505 0.0534 0.0618 0.0668 0.0714 0.0757 0.0798 0.0837 0.0874 0.0910 0.0944 0.0977 0.1009 0.1040 0.1070 0.1100
0.040 0.042 0.044 0.046 0.048 0.050 0.052 0.054 0.056 0.058 0,060 0.062 0.064 0.066 0.068 0.070 0.072 0.074 0.076
0.1128 0.1156 0.1183 0.1212 0.1236 0.1262 0.1287 0.1311 0.1335 0.1359 0.1382 0.1405 0.1423 0.1449 0.1471 0.1493 0.1514 0.1535 0.1555
0.078 0.080 0.10 0.12 0.14 0.16 0.18 0.20
0.1576 0.1596 0.1784 0.1954 0.2111 0.2256 0.2342 0.2520 0.2650 0.2756 0.2865 0.2968 0.3066 0.3159 0.3248 0.3333 0.3143 0.3489 0.3562
0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80
0.3631 0.3679 0.3760 0.3820 0.3877 0.3931 0.3981 0.4031 0.4078 0.4122 0.4165 0.4205 0.4242 0.4279 0.4314 0.4347 0.4409 0.4437 0.4470
0.22
0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42
Journal of Colloid and Interface Science, Vol. 28, No. 1, September t968
34
DENNIS TABLE II
GEL DIFFUSION
COEFFICIENTS
(×105
CM 2 GM) OF
~ I V E E L E C T R O L Y T E S IN T H E A , B , AND C GELS Gel
A B C Water
(25°C)
NaC1
KBr
KCI
CaCh
KI
2.2 2.0 2.2 1.5
2.9 3.1 2.9 2.0
2.8 2.7 2.7 1.9
1.9 2.7 1.9 1.2
2.9 2.8 3.1 2.1
subsequent D's can be averaged for a best value. Table II gives the gel diffusion coefficients (average of ten values, average deviation 4 - 5 % ) of five electrolytes in hydrochloric, tartaric, and acetic acid gels, referred to, respectively, as gels A, B, and C. The standard gel, hereafter referred to as the " S " gel, contained 0.125 gm/cc of Si02 from Na~Si03. 5H20. A constant hydrosol p H of 5.5 was maintained by mixing the neeessary amounts of acid with the silicate. It is seen that in the " S " gels diffusion coefficients, and hence diffusion rates, are greater than those pertaining to aqueous solution. This result rules out the possibility of reduced flow rates causing good growth of crystals in gels. I t seems likely that some other phenomenon such as the absence of convection currents is responsible. If Xfl is nearly constant for a particular gel, the equivalent statement, D g e l / D w ~ = constant, should hold. I t is easily seen that the ratio does not v a r y much for a particular gel. The subject of interest is not the ratio, but X, and so gels with fl = 1 are needed. When the silica concentration is increased to 1.5S, the ratio decreases to less than one. If it is assumed that surface diffusion becomes negligible (fl = 1) in such gels, then X can be obtained from gels of high silica content ( > 1.5S), and an extrapolation can be made to S gels. Figure 1 shows an experimentM result, an eneouraging extrapolation to almost 100 % pore volume for zero silica eoneentration. Each point on the graph is an average of pore space determined by each electrolyte in Table I. Agreement was good among the five values except for CaC12, whieh consistently gave X's 30 % higher than the other salts. Possibly the extrapolation is
not valid when the diffusing ions are doubly charged. I t is seen that the pore volume (which excludes the volume occupied by the adsorbed ions) comprises about 80 % of the S gel total volume. PORE SIZE DETERMINATION At this stage of the investigation, it must be re-emphasized that assumptions upon which the following technique is based are probably incomplete. Future invsetigations should entail a more thorough analysis. Assumptions necessary are (a) a regular, ideal, gel model, and (b) the presence initially of an adsorbed monolayer of hydrated sodium ions, which can be removed b y base exchange. The ideal gel will consist of long, uniform tubes parallel to each Cartesian direction. Each "pore" is the intersection of three channels, and will be considered as a sphere of radius r, the width of each channel. The distance between pores is d. If r >> d, the structure consists of small gel particles connected by thin threads. Consider a cube of gel i cm on a side, with n tubes along each direction. There are n 2 tubes on a side, and 3 n ~ in the unit cube. The total volume of pore and channel space is X = 3n2~rr 2 - -
2n3(4~Irr3).
[51
The second term on the right is the overlapping correction to the volume. Since there
•" G E L A LO
•
-.
0.~
o
B
I:I
C
-,%,,
0.7
~. ~ z~
w0.6
>04 o 0.3 02 0.1 0,5 SILICA
1.0 I;,5 CONCENTRATION
2D 2.5 IN S UNITS
5.0
FIG. 1. Average pore volume vs. silica concentration for the A, B, and C gels.
Journal of Colloid and Interface Science, Vol. 28, No. I, September 1968
35
DIFFUSION STUDIES IN SILICA GELS TABLE III PORE VOLL~iVfES~SIZES~ AND DENSITIES Gel
Xl
X2
nr
n(r - 5.1)
r(A)
d~)
n(X 10~)
A B C
0.82 0.77 0.80
0.97 0.97 0.97
0.385 0.385 0.385
0.360 0.345 0.354
79 49 65
47 30 39
4.9 7.7 5.9
are three rows, the volume of each intersection is accounted for three times, and therefore must be subtracted twice. Equation [5] yields the quantity (nr) when a numerical X is taken from Fig. 1. In order to determine separately n and r, their product must be determined from a gel with the monolayer removed. For this purpose, gels are base exchanged with 1% NH4C1 for one week, and then treated daily with fresh distilled water for one week, in order to remove the exchanged cations. In the treated gels the result is the determination of the quantity n(r - 5.1), 5.1 being the hydrated sodium ion diameter in Angstrom units. Finally, the pore radius and the linear density are calculated from the ratio n r / n ( r - 5.1). Table I I I lists the experimental results. The pore volumes before and after exchange, respectively, are X~and X2. Evidently the slight differences in effective pore space
are due to differences in the thic!mess of the adsorbed layer. The pore size of about 50 A may be compared with Plank's average value of 50 A for the dried gel (2). T E M P E R A T U R E D E P E N D E N C E OF T H E GEL D I F F U S I O N C O E F F I C I E N T
Initial work has revealed that the variation of the diffusion coefficient with temperature is not monotonic. As the temperature increases, surface diffusion decreases, but with further increase, the normal increase with temperature of the diffusion coefficient takes place. Thus the diffusion coefficient has a maximum value at some temperature. REFERENCES 1. DENNIS, J., To appear ill J . C h e m . E d u c . 45, 432-433 (1968). 2. PLANK, C. J., J . C o l l o i d S c i . 2,399 (1947).
Journal of Colloid and Interface Science, Vol. 28, No. 1, September 1968