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Discrete Charge Distributions in Dielectric Droplets
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
206, 19 –28 (1998)
CS985599
Discrete Charge Distributions in Dielectric Droplets M. Labowsky Department of Chemical Engineering Yale University, New Haven, Connecticut 06520 Received May 12, 1997; accepted April 20, 1998
the droplet (ion emission). Indeed, it is the ability to emit large, multiply charged macromolecular ions, such as proteins, that makes the electrospray such a valuable research tool. In light of this, it is important to understand the ion emission process from the electrospray droplets. There have been two electrospray ion emission mechanisms proposed in the literature. In his pioneering work on the electrospray, Dole (17, 18) suggested that charged droplets containing macromolecules go through a series of Rayleigh explosions. Eventually a point is reached when the droplets are small enough to contain only one macromolecule per droplet. The macromolecule is then charged when the solvent is completely evaporated, leaving a residual charge on the macromolecule. The other proposed ion formation mechanism is ion evaporation from a highly charged droplet. Previous studies on ion evaporation from charged droplets (19, 20) assumed a continuous charge distribution and focused on the fields existing outside the droplets. The rate of ion evaporation was determined based on the ability of an ion to penetrate a potential barrier, known as the Schottky Hump, after the ion left the surface of the droplet. In contrast to the previously cited studies, the present work takes a different approach. The charge distribution is treated as discrete and not continuous. Of particular interest are the microscopic fields in the immediate vicinity of the charges. This work was inspired by Fenn (5) who was interested in the geometric pattern of charge on the surface of a droplet and the nature of fields surrounding the charges, insisting that on a local scale these charges must be considered discrete rather than continuous. In an attempt to verify Fenn’s contention, the fields inside a droplet are calculated here by using a superposition technique. This allows for a rigorous calculation of the fields inside (as well as outside) the droplet. Knowing the fields, the electrostatic pressure exerted on the surface of the droplet can then be determined as a function of the number of charges and the dielectric constant of the droplet. The results obtained suggest a third possible ion formation mechanism, one in which a strong, localized, electrostatic pressure locally distorts the droplet surface eventually releasing an ion or small droplet.
The electrospray ion source has recently revolutionized mass spectrometry by allowing researchers to produce gaseous ions of very large molecular weights such as proteins and polymers. The process by which these high-molecular-weight ions are produced, however, is not very well understood. The purpose of the present work is to study the fields in the vicinity of each charge in order to shed some light on the ion formation process for charged dielectric droplets. An important feature of this work, therefore, is the treatment of the charge as discrete rather than continuous. The picture that emerges is of discrete charges in a dielectric droplet thermally oscillating around a potential well, located slightly below the droplet surface. The charges produce localized electrostatic pressures on the droplet surface that are higher (15.5 and 70% for dielectric constants of 80 and 20, respectively) than expected if the charge distribution were continuous. These high localized pressures could locally stretch the surface and result in the emission of ions or small charged droplets. The magnitude of these localized pressures is a function of the number of charges and the dielectric constant of the droplet. © 1998 Academic Press Key Words: electrospray; ion formation; charged droplets; dielectric droplets; discrete charge distributions.
INTRODUCTION
The subject of charged dielectric droplets is deserving of study in light of the recent, widespread use of the electrospray ion source (EIS) in mass spectrometry (1–5). In the electrospray process, solvent and solute molecules are forced through a needle, which is maintained at a high potential with respect to a grounded plate, and atomized. The resulting droplets are highly charged. With subsequent evaporation, the droplet size decreases, so the charge density increases. At some point, the charge density becomes sufficiently high that the droplet becomes unstable and loses charge. Charge loss occurs when the droplets either break into smaller, daughter, droplets (Rayleigh explosions)1 or when solute molecules are emitted as ions from 1 There have been numerous previous studies concerning the distortion, stability and break-up of charged droplets (6 –16). These studies involved the reaction of the droplet to the stress exerted on it by the resident charge. Common to all these studies was the assumption that the charge is continuously distributed on the droplet surface.
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0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
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M. LABOWSKY
METHOD OF SOLUTION
In order to evaluate the fields surrounding the multiple charges in a droplet with a dielectric constant, «,2 the fields near a single charge in a droplet must first be calculated. The solution to the single charge in a droplet problem can be found in Ref. (21). If the charge is located inside the droplet, the potential field inside the droplet is
potential is continuous, (b) the tangential component of the potential gradient is continuous, and (c) the radial component of the potential gradient is discontinuous. The radial component of the gradient outside the dielectric is equal to that inside the droplet times the dielectric constant. This is the solution for a single charge in a dielectric droplet. In light of the fact that the Laplace equation is linear, the solution for multiple charges in a droplet is constructed by simply superimposing the individual charge solutions
O ~n 1 1! `
v I 5 1/~«r c ! 1 ~« 2 1!/«
3 r n0 r n P n /@n ~« 1 1! 1 1# ,
V ~r! 5 [1a]
while the potential field outside the droplet is
O ~2n 1 1!r P /@~n ~« 1 3! 1 1!r `
vO 5
n 0
n
~n11!
#,
[1b]
n50
where r is the distance from the center of the droplet to the point of interest, r0 is the distance from the center of the droplet to the charge and rc is the distance from the charge to the point of interest. Distances are measured in units of droplet radii (a). Pn is the Legendre polynomial of order n. Implicit in using these equations is the fact that the droplet does not deform because it is in the presence of the charges. The limits of this assumption will be discussed later in this paper. The first term in Eq. [1a] represents the potential contribution of the point source inside the dielectric droplet. The summation terms in Eqs. [1a, b] represent the potential contribution arising from the polarization of the surface due to this point source.3 These equations have the properties that they are solutions to the Laplace equation and, on the surface of the droplet, they satisfy the conditions vI 5 vO ,
[2a]
dv I / dt 5 d v O / dt ,
[2b]
and
j
[3]
j51
where N is the number of charges in the droplet, V(r) is the potential field at vector position r and vj(r) is the contribution to the potential field at r resulting from the jth charge. Because the vj are solutions to the Laplace equation, V(r) is also a solution to this equation. Further, because each of the vj(r) individually satisfy the boundary conditions, V(r) will also satisfy the boundary conditions. Hence, V(r) represents the desired solution to the problem. The field strength at any point between the charges is found by taking the gradient of the potential E~r! 5 2=V ~r! , where = denotes the dimensionless gradient. The dimensionless total potential energy (TPE) for a charge distribution is defined as
O O ~u 1 I !/ 2 , N
TPE 5
N
ij
ij
[5]
i51 j51
where uij is u ij 5 1/~«~r i 2 r j !! if i Þ j 5 0 if i 5 j and
«dvI /dr 5 dvO /dr .
[2c]
These conditions state, that at the surface of the droplet (a) the 2
O v ~r ! , N
n50
In this work, dielectric constant refers to the relative dielectric constant, defined as the ratio of the dielectric constant of the droplet to that of the ambience. The dielectric constant of the ambience is the same as in air (vacuum). 3 Note that a similar set of equations can be written for the case of a charge outside a dielectric droplet.
O ~n 1 1!r r P /@n ~« 1 1! 1 1# . `
I ij 5 ~1 2 1/« !
n n i j
n
n50
In these expressions, ri is the position vector of the ith charge. The magnitude of ri is ri. The TPE is the total of the potentials at each of the charge sites, excluding singular terms. The factor of 2 in the denominator is included to eliminate double counting.
21
DISCRETE CHARGE DISTRIBUTIONS
The work (W) required to move a charge is calculated by integrating the field strength over the path traced by the charge
E
W 5 2 =V z ds ,
[6]
where ds is the displacement vector along the path of motion. Note that these equations have been normalized so distances are measured in units of the droplet radius (a). The potential field is normalized so that the potential a unit distance (a) from a charge of unit strength is unity. To obtain the potential in mks units (9), the dimensionless potential (V) calculated in Eq. [3] must be multiplied by a normalizing term 9 5 Ve /~4 p « 0 a ! ,
[7]
where e is charge on an electron and «0 is the electric permittivity of vacuum. The mks field strength (%) is related to the dimensionless field’s strength by % 5 eE /~4 p « 0 a 2 ! .
[8]
Similarly, the mks total potential energy of the distribution ((3%) and mks work (0) are related to their dimensionless cousins by (3% 5 TPE e 2 /~4 p « 0 a !
[9]
0 5 We 2 /~4 p « 0 a ! .
[10]
and
All that remains in solving this problem is to specify the distribution of the charge in the dielectric. It is conventional wisdom that charge will distribute itself ‘‘uniformly’’ on the surface of a spherical conductor. ‘‘Uniformity’’ of charge distribution is a concept that cannot be applied to a discrete number of charges. For example, how does one place five charges, ‘‘uniformly’’ (i.e., so one charge is indistinguishable from the others) on the surface of a sphere? It is not possible. In fact, charges do not distribute ‘‘uniformly’’ but rather in a distribution in which the total potential energy (TPE) is a minimum. In the present study, the charge distributions were determined by first placing the charges randomly inside the dielectric droplet and then allowing them to rearrange in response the forces exerted on them until the minimum TPE distribution is attained.
FIG. 1. The field lines in the vicinity of one charge located at a radial position of 0.943 in a droplet with 100 charges and a dielectric constant of 80.
RESULTS
Using the preceding approach, it is straightforward to calculate the fields surrounding the charges in a dielectric droplet. Figure 1 shows the field lines near a charge in a droplet containing 100 charges and with an « 5 80. In many of the examples presented in this paper, the dielectric constant of 80 will be used because the « for water is about 80. The charge is located 0.057 radii from the surface. Notice the discontinuities in the field lines at the surface of the droplet. These discontinuities result from the jump boundary condition in the normal component of the potential gradient at the surface of the droplet (Eq. [2c]). Physically, these discontinuities reflect the polarization surface charge arising from the charges inside the dielectric. It is evident from the examination of Fig. 1 that there is a polarization charge induced on the surface of the droplet. A single charge in an infinite medium will be uniformly surrounded by dielectric material and will feel no net force. If that same charge were located near the inner surface of a dielectric droplet, the charge would induce a polarization charge on the surface of the droplet. This polarization charge would exert a force on the original charge, pushing it to the center of the droplet. This force will be referred to as the ‘‘polarization force.’’ A single charge in a dielectric droplet will tend to move to the center of the droplet because of this polarization force. If a second charge were added to the droplet, the charges would exert repulsive forces on one another, which would tend to push them apart, towards the surface of the droplet. This repulsive force is referred to as the interaction force because it arises from the coloumbic interaction between the charges. So, the outward interaction force is opposed by the inward polarization force. The charges locate where the interaction force balances the polarization force. The exact position of this equilibrium point depends on the value of the « and on the number of charges (N) in the droplet. If « were unity, there would be no polarization force, and the charges would be
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M. LABOWSKY
charges, one test charge is moved, while the other N 2 1 charges are fixed in their otherwise equilibrium position. The change in the work (W) required to move the test charge from the center of the droplet to the surface is then calculated and compared with the thermal energy of the charge. Figure 3a shows the work as a function of radial distance (R) for the case of N 5 100 and « 5 80. The work is assigned a value of zero
FIG. 2. The equilibrium sphere of charge radius (Req) as a function of the number of charges (N) and the dielectric constant.
forced to the surface of the droplet by the interaction force. As « increases, the polarization force also increases. The polarization force pushes the charges away from the surface. As mentioned previously, the equilibrium position of the charges is determined by allowing the charges to move in response to the forces acting on them until the TPE is a minimum. In so doing, it was found that the charges arrange approximately on a ‘‘sphere of charge’’ within the droplet.4 The radius of this ‘‘sphere of charge’’ is less than the radius of the droplet. Figure 2 shows the radius (Req) for the equilibrium sphere of charge as a function of the number of charges (N) on the droplet. This figure has five curves, each representing a different value of «. Figure 2 clearly shows the interplay between the interaction and the polarization forces. As the number of charges increases for a fixed «, the equilibrium position moves closer to the surface of the droplet because the outward interaction force increases with increasing N. Alternatively, the equilibrium radius decreases as « increases for a given N because the inward polarization force increases with increasing «. It is noteworthy that for « larger than 10, the equilibrium position is fairly insensitive to «. The curves for all the values of « approach the limiting case of 1 for an infinite amount of charge on the droplet. Figure 2 shows the equilibrium positions of the charges on the sphere of charge. The question as to how firmly the charges are locked into that equilibrium position then arises. For example, all the charges will have associated with them a certain amount of thermal energy. How far can the charges move as a result of this thermal motion? To answer this question, one charge in the distribution will be designated as a ‘‘test’’ charge. This test charge is allowed to move in the radial direction, while all other charges are locked on the equilibrium sphere of charge. To clarify this point, if a distribution contains N 4
Except in a few cases, the charges do not locate exactly on a sphere of charge. In general, charges arrange slightly above and slightly below the sphere of charge.
FIG. 3. (a) Work as a function of radial position for a test charge in a droplet with N 5 100 and « 5 80. (b) Work as a function of radial position and « for a droplet with N 5 100. (c) Work as a function of radial position and « for a droplet with N 5 200.
DISCRETE CHARGE DISTRIBUTIONS
at the center of the droplet. As the test charge moves from the center, W is negative. This means that the charge experiences an outward interaction force which would push it from the center toward the surface. The slope of the curve is small near the center but becomes steeper (more negative) as R increases. This means that the outward interaction force acting on the charge increases with R. In the region near R 5 0.943, a potential well is reached. This potential well corresponds to the equilibrium sphere of charge point found in Fig. 2. Beyond this well, the slope of the work curve changes and becomes positive. The test charge experiences a net inward force because the inward polarization forces are larger than the competing outward interaction forces. This polarization force would become very large (infinite) if the charge were able to get infinitesimally close to the surface. The net inward force becomes stronger the closer the charge moves to the surface. Indeed, in the classical sense, the surface represents an impregnable potential barrier to the charge. Figure 3b shows the work curves for several values of « for a droplet with 100 charges. The depth of the potential well increases as the dielectric constant decreases. For small «, the outward coloumbic interaction force is more pronounced near the center of the droplet than for large «. However, even for small «, there is a strong inward polarization force when the test charge is very near the surface. The potential well is a stable equilibrium point for the charge. If the charge is perturbed from the equilibrium position, it will tend to return to that position. This is in contrast to the ‘‘Schottky Hump’’ (19, 20) located outside the droplet which is an unstable equilibrium point. Figure 3c shows the work curves for several value of « for the case of a droplet with 200 charges. The curves are qualitatively the same as those shown in Fig. 3b; however, the potential wells are deeper. The depth of the potential wells reflects the increased strengths of the interaction and polarization forces as the number of charges increases. The dependence of the depth of the potential well on N for various « is shown in Fig. 4. An examination of this figure clearly shows that as N increases, or as « decreases, the depth of the potential well increases. For droplets with small values of «, the potential wells can be very deep. Up to this point, only dimensionless quantities have been used in order to understand qualitative trends. It is now of interest to see quantitatively how tightly the charges are held to the equilibrium point. The charges will have a certain amount of thermal energy. It is important to compare this thermal energy with the depth of the potential well surrounding the charges. Consider a droplet which has a dielectric constant 80, a radius of 100 angstroms (Å), and 100 charges. In this case, the bottom of the potential well is at 0.943, or a distance of 5.7 Å from the surface of the droplet (see Fig. 3b). The depth of the potential well is 0.168 which corresponds to an energy of 3.87 3 10214 ergs (see Eq. [9]). For a droplet at 300 K, the
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FIG. 4. The potential well depth as a function of the number of charges (N) and «.
charge would have 3kT/2 or 6.21 3 10214 ergs of thermal energy (TE). Hence, the ratio of the TE to the potential well depth is 1.6. This means that if the charge has 3kT/2 of TE, it could move to the center of the droplet and could approach the surface to a radial position of about 0.99. So, thermal motion would allow the charges to move freely about the equilibrium position. If the droplet had a radius of 50 Å, the ratio of TE to the well depth would be 0.8. The charge would be able to move only between radial positions about 0.5 and 0.985. Conversely, if the droplet had a radius of 200 Å, the TE to well depth ratio would be 3.21. The charge would be able to reach the center of the droplet and could reach a radial position greater than 0.995. It is clear from this that charges would be able to thermally move more freely in larger rather than in a smaller droplets if the larger and smaller droplets had the same amount of charge. If we looked at the case in which « were 10, the potential well is noticeably deeper. For 100 charges, the well is located at 0.947 and has a depth of 1.286 (see Fig. 3b). For a droplet with a 100-Å radius, the well locates 5.3 Å from the surface and has a depth of 2.96 3 10213 ergs. The TE to well depth ratio is 0.21, indicating that the charges are very tightly bound to the potential well. The charge would be able to move only between the radial positions of about 0.86 and 0.975, so the thermal motion is significantly restricted. The same dependence on droplet size as in the « 5 80 case would be found here. The thermal motion of the charge would be more restricted in smaller droplets than in larger ones. In the preceding calculation, only one charge was moved while the others remained locked in their equilibrium positions. The purpose was simply to show the magnitude of the potential well near each charge. If a charge in a droplet were to move in response to this disturbance, and the calculated work would be expected to be different from that calculated here. The response to the disturbance, however, will be transient and depend on the mobility of the ions in the dielectric. It is also clear from these calculations that for large « the charges will not statically locate at a specific point but will dynamically move
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M. LABOWSKY
as a result of thermal motion. The dynamics of charge motion in dielectric droplets is a fertile area for research but one which is beyond the scope of the present paper. Up to this point, we have been concerned with the forces acting on a test charge as it moves from its equilibrium position. As has been seen, the surface polarization force acts to push the charge toward the center of the droplet. However, as the surface exerts an inward force on the charge, the charge exerts an outward force on the surface. It is important to study this force because it may play a crucial role in the ion emission process. The expression for the electrostatic pressure (3) (referred to henceforth simply as the pressure), acting on the surface of dielectric, can be found in any number of textbooks (see, for example, Ref. 22) 3 5 « 0 ~« 2 1!@% 2t 1 «% 2n #/ 2 ,
[11]
where %t and %n are, respectively, the tangential and the normal components of the electrostatic field strengths on the dielectric side of the surface. If all N charges were located at the center of the droplet, then %n 5 eN/(4p«0a2)«, %t 5 0, and the pressure would be that for a continuous charge distribution 3 cont 5 ~« 2 1!@e 2 N 2 /~4 p a 2 ! 2 «« 0 #/ 2 .
[12]
This pressure would be uniform over the entire droplet surface. 3cont serves as a useful reference pressure. It is the pressure which would be exerted on the surface of a droplet if the charge were continuously distributed on the surface of the droplet. Note that 3cont is directly proportional to N2 and has a weak dependence on « except when « is small. A dimensionless pressure (P) is defined as the actual pressure (3) divided by 3cont P 5 @E 2t 1 «E 2n #/@N 2 /« # ,
FIG. 5. The electrostatic pressure (P) exerted on the surface of a droplet in the vicinity of two adjacent charges in their equilibrium positions for a droplet with N 5 100 and « 5 80.
[13]
where Et and En are the dimensionless field strengths inside the droplet in the tangential and normal directions, respectively. This dimensionless pressure is a direct measure of the uniformity of the actual pressure. If P is unity, then the pressure is the same as if the charge were continuously distributed on the droplet. Deviations from unity indicate that the charge distribution is not continuous. In the following, the term pressure will be used to describe the dimensionless electrostatic pressure. Figure 5 shows the pressure, as a function of the fractional angular displacement (Q/Q0) between a test charge and an adjacent charge for the case of « 5 80 and N 5 100 when both charges are at their equilibrium positions (R 5 0.943). Q0 is the angle between the lines joining the adjacent charges to the center of the droplet. Q/Q0 is 0 above the test charge and 1
above the adjacent charge. On the surface above the test charge (Q/Q0 5 0), the pressure is 1.137. This means that on the surface immediately above the test charge, the pressure is 13.7% higher than that for a continuous distribution. As Q/Q0 increases, P gradually rises to a maximum (PMAX) of 1.15 at a Q/Q0 value of 0.06. P decreases beyond this maximum and reaches a minimum value of 0.99 at the midpoint (Q/Q0 5 0.5) between the two charges. This minimum is slightly below unity, indicating that the pressure is below the continuous charge distribution pressure. The pressure curve is symmetrical about the mid point and the same pressure hump is observed over the adjacent charge as was observed over the test charge for this position of the test charge. Figure 6a shows the pressure as the test charge is moved from its equilibrium position and the other charges remain locked in their equilibrium positions. As the test charge approaches the surface, the pressure noticeably increases. For R 5 0.95, PMAX is 1.22. For R 5 0.96, PMAX is 1.45. For R 5 0.97, PMAX is 2.19. So, small deviations from the equilibrium position of the test charge can result in very large deviations of the local pressure from the continuous distribution pressure. This local pressure can be several times larger than that for a continuous distribution. As was mentioned earlier, if the test charge had 3kT/2 of thermal energy, it could easily reach a radial position of R 5 0.97 even for 50-Å droplets with « 5 80. If the test charge moved to a radial position of 0.98, the maximum pressure would be more than 5 times greater than assumed using a continuous charge distribution. Note that while the pressure in the immediate vicinity of the test charge increases dramatically, this pressure disturbance decreases rapidly from the maximum. For example, for R 5 0.97 the maximum is 2.19 at Q/Q0 5 0.06, but it decreases to 1.28 at Q/Q0 5 0.15. This means that the large pressure disturbances are highly localized. The size of the highly stressed surface area can be estimated. For N 5 100, the
DISCRETE CHARGE DISTRIBUTIONS
distance between charges is of the order of 0.355 radii. A Q/Q0 of 0.15 then corresponds to a disturbance diameter of 0.106 radii or 10.6 Å for a 100-Å droplet. A Q/Q0 of 0.06 corresponds to a pressure disturbance diameter of only 0.042 radii or 4.2 Å. So, the high-pressure areas on the surface are very localized. It should also be mentioned that when the pressure above the test charge is significantly larger than the continuous distribution pressure, the pressure at the midpoint between the charges droplet is slightly below the continuous pressure. The pressure above the adjacent charge, on the other hand, is nearly the same as that as when the test charge is in the equilibrium position. Figure 6b shows the pressure curves for the case of N 5 200. It is interesting that, when the test charge is in its equilibrium position (R 5 0.96), the curve is similar to the equilibrium curve for N 5 100. As the test charge approaches the surface, however, the rise in the maximum value of P is not as great as that for N 5 100. As may be seen, PMAX is 1.37 when R 5 0.97 compared with 2.19 for the N 5 100 case. This is significant because it indicates that PMAX decreases with increasing N for
FIG. 6. The electrostatic pressure (P) exerted on the surface between a test charge and an adjacent charge as a function of the radial position of the test charge (a) for the case of a droplet with N 5 100 charges and « 5 80 and (b) for the case of N 5 200 and « 5 80.
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FIG. 7. The maximum electrostatic pressure (PMAX) exerted on the surface as a function of the number of charges and the dielectric constant for a test charge position of 0.96.
a given test charge position.5 It should also be noted that when the test charge is below its equilibrium point (R 5 0.95 curve), the value of PMAX approaches the continuous limit of unity. This would be expected because continuous distribution approximation is better the farther a charge is from the surface. The important point to be made here is that even though the pressure approaches the continuous distribution limit of unity when the charge is far from the surface, PMAX is 15% greater than predicted by a continuous charge distribution assumption when the charge is in its equilibrium position. The dependence of PMAX on N for test charge in a fixed radial position is shown in Fig. 7. This figure is a graph of PMAX as a function of N with five curves for each of the indicated values of «. This figure is for a test charge at a radial position of 0.96. All the curves show that as the number of charges increases, PMAX decreases and asymptotically approaches the continuous distribution P limit of unity. The figure is broken into two regions labeled ‘‘above’’ and ‘‘below’’ and separated by a broken line. The purpose is to indicate whether the test charge at a radial position of 0.96 is above or below its equilibrium position. For example, the equilibrium position for a distribution with 500 charges with an « of 80 is 0.975, so a test charge position of 0.96 is below the equilibrium value. PMAX can be very large for small values of N. For N 5 50 and « 5 80, PMAX is 2.5. For N 5 50 and « 5 40, PMAX is 4.0. So, for small values of N, PMAX can be several times larger than that expected by assuming a continuous charge distribution. As N increases, PMAX decreases rapidly at a rate which is inversely proportional to N2. The approach to a continuum distribution for large N evident in Fig. 7 is somewhat misleading. In this figure, the reason the Remember that PMAX is normalized by 3cont, which directly depends on N . Consequently, while the dimensionless pressure decreases with N, the magnitude of the actual pressure increases as a result of this square dependence on N. 5
2
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M. LABOWSKY
FIG. 8. The maximum electrostatic pressure (PMAX) exerted on the surface when all charges are in the equilibrium positions as a function of the number of charges and the dielectric constant.
continuum limit is approached is that by keeping R fixed the charges locate below their equilibrium positions when N is large. The deeper a charge is buried in the droplet, the more continuous it appears to the surface. Conversely, in Fig. 7, PMAX is large when N is small because the charges are above their equilibrium positions when N is small. So Fig. 7 shows PMAX values that tend to be too low for large N and are too high for low N. In contrast, Fig. 8 shows PMAX when the test charge is at its equilibrium position (PMAX, EQ). This figure reveals that PMAX, EQ is only a weak function N. For « 5 80, PMAX, EQ is 1.145 for N 5 50 and 1.157 for N 5 500. This means that for N 5 50 and N 5 500, PMAX, EQ is, respectively, 14.5% and 15.7% higher than would be predicted based on an assumption of a continuous charge distribution. It is significant that PMAX, EQ increases slightly when N changes from 50 to 500. The fact that PMAX, EQ is nearly independent of N means that the values of PMAX, EQ calculated here will persist and may increase slightly for extremely large values of N. It is also clear from Fig. 8 that as « decreases, the value of PMAX, EQ increases significantly. For example, for N 5 500 and « 5 40, PMAX, EQ is 1.3. For « 5 20, it is 1.7. For « 5 10 and 5, PMAX, EQ is, respectively, 2.46 and 5.1. All this indicates that the assumption of a continuous charge distribution to describe the pressure on the surface of a charged droplet is basically flawed. Even though the error associated with a continuous charge distribution may be acceptable when « is large (on the order of 15% for « of 80), the actual value of PMAX, EQ will always be larger than that predicted by a continuous charge distribution regardless of the number of charges on the dielectric droplet. DISCUSSION
As mentioned in the introduction, charge is lost by either the emission of charged daughter droplets or by the emission of ions. The present work has shown only that the surface above
charges in dielectric droplets can be locally very highly stressed. The response of the surface to that stress will be the subject of future work. It is appropriate here, however, to discuss the possible implications of the present results. Rayleigh derived an instability limit by considering the perturbation of a droplet which was initially spherical and had a continuous charge distribution. If the perturbed droplet returned to its spherical shape, it was stable, and if it did not, the droplet was unstable and would lose mass and charge. Rayleigh did not discuss the nature of the perturbation only that the perturbation existed (6). A charged droplet can be perturbed from its spherical shape in any one of a number of ways. There may be hydrodynamic forces acting on the surface. Perturbations may also result when the droplet is in a nonuniform electric field because of external field gradients, encounters with other charged droplets, or statistical fluctuations of the charge distribution. Such perturbations would disturb the spherical symmetry of the droplet and/or its charge and could cause a highly charged droplet to become unstable and lose charge. But suppose the charged droplet were in a perfectly quiescent and uniform environment so there are no external hydrodynamic or electric effects. If the droplet were considered to have a perfectly continuous charge distribution, the charge would exert a perfectly uniform pressure on the surface of the droplet. In this highly idealized situation, there would be no obvious site where an instability would occur. The importance of the present work is to show the charge distribution cannot be considered continuous. As may be seen from Fig. 8, even for very large values of N, there are nonuniform local pressures on the surface in the vicinity of the charges which can perturb the surface. In the absence of external effects, the surface immediately above any of the charges would be a potential site of instability. Further, the Raleigh instability limit which was derived assuming a continuous charge distribution may require modification in order to account for the local nonuniform pressure in the vicinity of the charges. These results allow us to make some interesting conjectures concerning ion emission from dielectric droplets. If a charge were to move infinitesimally close to the surface, it would hit an infinitely high potential wall. In a classical sense, the charge would not be able to penetrate this potential wall and would be trapped inside the droplet. In practice, however, the charge has a finite size and so cannot move infinitesimally close to the surface. Further, as the surface charge exerts a force on the charge, the charge exerts an equal but opposite force on the surface. The droplet surface will tend to deform in response to this force. The amount of deformation will depend on the ratio of the electrostatic pressure exerted on the surface by the charge to the restoring surface tension on the droplet. If this pressure ratio is small, there will be little deformation of the surface. If the pressure ratio is large, the surface will deform. As was shown earlier, the local pressure exerted on the surface can be large. This local pressure would exist even if the
DISCRETE CHARGE DISTRIBUTIONS
27
FIG. 9. The proposed mechanism for charge loss. (a) The charge approaches the surface. (b, c) The force exerted on the surface because of the charge distorts the surface and draws the charge outward. (d) The distorted surface separates from the droplet forming a charged ion. (e) If the force is slowly exerted on the surface, the surface may deform into a minicone.
charges were in their equilibrium positions but would become more pronounced if the charges moved toward the surface as a result of thermal motion. The pressure nonuniformity could be several times greater than that for a continuous distribution. This means that there can be a local stretching of the surface with the result that small bumps will tend to form in the vicinity of the charges (see Fig. 9a). These bumps may occur as a result of either a ‘‘one shot’’ encounter with a charge or a resonance induced by repeated encounters with a charge which
is oscillating about its equilibrium position. As was seen previously, the size of the bumps would be much of the size of a solvated ion. It is speculated that these bumps enlarge (Fig. 9b). As they enlarge, the surface moves away from the charge, allowing the charge to be drawn toward the surface. At this point, one of two possibilities emerge. Either the bump expands rapidly and detaches from the surface (Fig. 9c, d), encapsulating the charge and liberating a (solvated) ion, or the bump expands slowly and stretches the surface to form a
28
M. LABOWSKY
microscopic cone, pulling in several neighboring charges (Fig. 9e). Roellgen (23, 24) has suggested that ion evaporation is not likely to occur and is critical of the work of Iribane and Thomson (I&T) (19) claiming that solvated ions are emitted when a Raleigh instability somehow degenerates into a series of microscopic cones. Roellgen does not, however, propose a mechanism for this process. The present work may point to such a mechanism, although confirmation must await future calculations. The present work focused only on the electrostatic fields near discrete charges. Several physical properties will also play a role in the ion emission process. One such property is the size of the ion. High electrostatic pressures are found when a charge is very close to the surface. A large ion may not physically be able to get sufficiently close to exert these large pressures on the surface. This and other considerations will be the subject of future work. This future work may follow along the lines of the calculations of Basaran (11–15). Basaran has provided detailed calculations of the large-scale deformation of liquid droplets and liquid in tubes in response to electrical fields. Basaran used continuous charge distributions and boundary integral methods in his work. The future work will attempt to incorporate discrete charge distributions into the boundary integral methods in an attempt to see how the droplet surface responds to the localized electrostatic pressure near each charge. When this future work is completed, comparison of the theory with experiments will be possible. CONCLUSIONS
This paper presents a method for calculating the electrostatic properties of dielectric droplets with discreet charge distributions. Several interesting observations can be made from the results. Charges in a dielectric droplet will tend to locate slightly below the surface of the droplet in a position in which the inwardly directed polarization force equals the outwardly repulsive force of the adjacent charges. The electrostatic pressure exerted on the surface of the droplet is highly localized above a charge. This pressure can be extremely high if the charge strays toward the surface from its equilibrium position. The fact that the local pressure can be very high and the charges are located slightly below the surface leads to the conjecture of an ion emission mechanism in which an ion pierces the surface of the droplet. Further, the fact that the electrostatic pressure near a charge will always be higher than that calculated using a continuous charge distribution means that the classical Rayleigh instability limit, which is based on a continuous charge distribution, may require modification. Clearly more work needs to be done in this area. The present work, in fact, has raised more questions than it has answered.
However, the questions that have been raised are important and deserving of further study. The next phase of this research would be to examine the dynamics of the charge interactions and to model the local stretching of the surface by the charges with the intent of calculating the rates at which singly charged ions are emitted from the droplets. It would also be important to see if local stretching is a sufficient disturbance to initiate a large-scale instability for droplets near the Rayleigh limit. In conjunction with this next phase of research, it would be important to study the dynamics of charges which are bound to neighboring charges with the hope of better understanding the multiply charged ion emission process and to see if and when there is preferential ion emission for multiply charged ions. ACKNOWLEDGMENTS The author thanks Prof. J. B. Fenn of Virginia Commonwealth University and Prof. Juan Fernandez de la Mora of Yale University for their many thoughtful discussions, insights into this problem, and help in preparing this manuscript. The author also thanks Anna and Sarah Labowsky for their help in preparing the graphs.
REFERENCES 1. Labowsky, M. J., Fenn, J. B., and Yamashita, M., U.S. Patent # 4,531,056 (1985). 2. Meng, C. K., Mann, M., and Fenn, J. B., Z. Phys. D 10, 361 (1988). 3. Fenn, J. B., et al., Science 246, 64 (1989). 4. Fenn, J. B., et al., Mass Spectom. Rev. 9, 37 (1990). 5. Fenn, J. B., J. Am. Soc. Mass Spectrom. 4, 524 (1993). 6. Rayleigh, Lord, Philos. Mag. 14, 184 (1882). 7. Ryce, S. A., and Wyman, R. R., Can. J. Phys. 42, 2185 (1964). 8. Taylor, G. I., Proc. R. Soc. London Ser. A 280, 383 (1964). 9. Ryce, S. A., and Patriarche, D. A., Can. J. Phys. 43, 2192 (1965). 10. Milkis, M., Phys. Fluids 24, 1967 (1981). 11. Basaran, O. A., and Scriven, L. E., Phys. Fluids A 1, 795 (1989). 12. Basaran, O. A., and Scriven, L. E., Phys. Fluids A 1, 799 (1989). 13. Haywood, R. J., et al., AIChE J. 37, 1305 (1991). 14. Basaran, O. A., et al., Ind. Eng. Chem. Res. 34, 3454 (1995). 15. Harris, M. T., and Basaran, O. A., J. Colloid Interface Sci. 170, 308 (1995). 16. Fernandez de la Mora, J., J. Colloid Interface Sci. 178, 209 (1996). 17. Dole, M., et al., J. Chem. Phys. 49, 2240 (1968). 18. Mach, L. L., et al., J. Chem. Phys. 52, 4977 (1970). 19. Iribane, J. V., and Thomson, B. A., J. Chem. Phys. 64, 2287 (1976). 20. Loscertales, I. G., and De la Mora, J. F., J. Chem. Phys. 103, 5041 (1995). 21. Budak, B. M., Samarskii, A. A., and Tikhonov, A. N., ‘‘A Collection of Problems in Mathematical Physics,’’ pp. 480 – 481, Dover, New York, 1964. 22. Smythe, W. R., ‘‘Static and Dynamic Electricity,’’ pp. 18 –20, McGrawHill, New York, 1968. 23. Roellgen, F. W., Bramer-Weger, E., and Butfering, L., J. Phys. 48, C6-253 (1987). 24. Schmelzeisen-Redeker, G., Butfering, L., and Roellgen, F. W., Int. J. Mass Spectrom. Ion Proc. 90, 139 (1989).
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CS985599
Discrete Charge Distributions in Dielectric Droplets M. Labowsky Department of Chemical Engineering Yale University, New Haven, Connecticut 06520 Received May 12, 1997; accepted April 20, 1998
the droplet (ion emission). Indeed, it is the ability to emit large, multiply charged macromolecular ions, such as proteins, that makes the electrospray such a valuable research tool. In light of this, it is important to understand the ion emission process from the electrospray droplets. There have been two electrospray ion emission mechanisms proposed in the literature. In his pioneering work on the electrospray, Dole (17, 18) suggested that charged droplets containing macromolecules go through a series of Rayleigh explosions. Eventually a point is reached when the droplets are small enough to contain only one macromolecule per droplet. The macromolecule is then charged when the solvent is completely evaporated, leaving a residual charge on the macromolecule. The other proposed ion formation mechanism is ion evaporation from a highly charged droplet. Previous studies on ion evaporation from charged droplets (19, 20) assumed a continuous charge distribution and focused on the fields existing outside the droplets. The rate of ion evaporation was determined based on the ability of an ion to penetrate a potential barrier, known as the Schottky Hump, after the ion left the surface of the droplet. In contrast to the previously cited studies, the present work takes a different approach. The charge distribution is treated as discrete and not continuous. Of particular interest are the microscopic fields in the immediate vicinity of the charges. This work was inspired by Fenn (5) who was interested in the geometric pattern of charge on the surface of a droplet and the nature of fields surrounding the charges, insisting that on a local scale these charges must be considered discrete rather than continuous. In an attempt to verify Fenn’s contention, the fields inside a droplet are calculated here by using a superposition technique. This allows for a rigorous calculation of the fields inside (as well as outside) the droplet. Knowing the fields, the electrostatic pressure exerted on the surface of the droplet can then be determined as a function of the number of charges and the dielectric constant of the droplet. The results obtained suggest a third possible ion formation mechanism, one in which a strong, localized, electrostatic pressure locally distorts the droplet surface eventually releasing an ion or small droplet.
The electrospray ion source has recently revolutionized mass spectrometry by allowing researchers to produce gaseous ions of very large molecular weights such as proteins and polymers. The process by which these high-molecular-weight ions are produced, however, is not very well understood. The purpose of the present work is to study the fields in the vicinity of each charge in order to shed some light on the ion formation process for charged dielectric droplets. An important feature of this work, therefore, is the treatment of the charge as discrete rather than continuous. The picture that emerges is of discrete charges in a dielectric droplet thermally oscillating around a potential well, located slightly below the droplet surface. The charges produce localized electrostatic pressures on the droplet surface that are higher (15.5 and 70% for dielectric constants of 80 and 20, respectively) than expected if the charge distribution were continuous. These high localized pressures could locally stretch the surface and result in the emission of ions or small charged droplets. The magnitude of these localized pressures is a function of the number of charges and the dielectric constant of the droplet. © 1998 Academic Press Key Words: electrospray; ion formation; charged droplets; dielectric droplets; discrete charge distributions.
INTRODUCTION
The subject of charged dielectric droplets is deserving of study in light of the recent, widespread use of the electrospray ion source (EIS) in mass spectrometry (1–5). In the electrospray process, solvent and solute molecules are forced through a needle, which is maintained at a high potential with respect to a grounded plate, and atomized. The resulting droplets are highly charged. With subsequent evaporation, the droplet size decreases, so the charge density increases. At some point, the charge density becomes sufficiently high that the droplet becomes unstable and loses charge. Charge loss occurs when the droplets either break into smaller, daughter, droplets (Rayleigh explosions)1 or when solute molecules are emitted as ions from 1 There have been numerous previous studies concerning the distortion, stability and break-up of charged droplets (6 –16). These studies involved the reaction of the droplet to the stress exerted on it by the resident charge. Common to all these studies was the assumption that the charge is continuously distributed on the droplet surface.
19
0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
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M. LABOWSKY
METHOD OF SOLUTION
In order to evaluate the fields surrounding the multiple charges in a droplet with a dielectric constant, «,2 the fields near a single charge in a droplet must first be calculated. The solution to the single charge in a droplet problem can be found in Ref. (21). If the charge is located inside the droplet, the potential field inside the droplet is
potential is continuous, (b) the tangential component of the potential gradient is continuous, and (c) the radial component of the potential gradient is discontinuous. The radial component of the gradient outside the dielectric is equal to that inside the droplet times the dielectric constant. This is the solution for a single charge in a dielectric droplet. In light of the fact that the Laplace equation is linear, the solution for multiple charges in a droplet is constructed by simply superimposing the individual charge solutions
O ~n 1 1! `
v I 5 1/~«r c ! 1 ~« 2 1!/«
3 r n0 r n P n /@n ~« 1 1! 1 1# ,
V ~r! 5 [1a]
while the potential field outside the droplet is
O ~2n 1 1!r P /@~n ~« 1 3! 1 1!r `
vO 5
n 0
n
~n11!
#,
[1b]
n50
where r is the distance from the center of the droplet to the point of interest, r0 is the distance from the center of the droplet to the charge and rc is the distance from the charge to the point of interest. Distances are measured in units of droplet radii (a). Pn is the Legendre polynomial of order n. Implicit in using these equations is the fact that the droplet does not deform because it is in the presence of the charges. The limits of this assumption will be discussed later in this paper. The first term in Eq. [1a] represents the potential contribution of the point source inside the dielectric droplet. The summation terms in Eqs. [1a, b] represent the potential contribution arising from the polarization of the surface due to this point source.3 These equations have the properties that they are solutions to the Laplace equation and, on the surface of the droplet, they satisfy the conditions vI 5 vO ,
[2a]
dv I / dt 5 d v O / dt ,
[2b]
and
j
[3]
j51
where N is the number of charges in the droplet, V(r) is the potential field at vector position r and vj(r) is the contribution to the potential field at r resulting from the jth charge. Because the vj are solutions to the Laplace equation, V(r) is also a solution to this equation. Further, because each of the vj(r) individually satisfy the boundary conditions, V(r) will also satisfy the boundary conditions. Hence, V(r) represents the desired solution to the problem. The field strength at any point between the charges is found by taking the gradient of the potential E~r! 5 2=V ~r! , where = denotes the dimensionless gradient. The dimensionless total potential energy (TPE) for a charge distribution is defined as
O O ~u 1 I !/ 2 , N
TPE 5
N
ij
ij
[5]
i51 j51
where uij is u ij 5 1/~«~r i 2 r j !! if i Þ j 5 0 if i 5 j and
«dvI /dr 5 dvO /dr .
[2c]
These conditions state, that at the surface of the droplet (a) the 2
O v ~r ! , N
n50
In this work, dielectric constant refers to the relative dielectric constant, defined as the ratio of the dielectric constant of the droplet to that of the ambience. The dielectric constant of the ambience is the same as in air (vacuum). 3 Note that a similar set of equations can be written for the case of a charge outside a dielectric droplet.
O ~n 1 1!r r P /@n ~« 1 1! 1 1# . `
I ij 5 ~1 2 1/« !
n n i j
n
n50
In these expressions, ri is the position vector of the ith charge. The magnitude of ri is ri. The TPE is the total of the potentials at each of the charge sites, excluding singular terms. The factor of 2 in the denominator is included to eliminate double counting.
21
DISCRETE CHARGE DISTRIBUTIONS
The work (W) required to move a charge is calculated by integrating the field strength over the path traced by the charge
E
W 5 2 =V z ds ,
[6]
where ds is the displacement vector along the path of motion. Note that these equations have been normalized so distances are measured in units of the droplet radius (a). The potential field is normalized so that the potential a unit distance (a) from a charge of unit strength is unity. To obtain the potential in mks units (9), the dimensionless potential (V) calculated in Eq. [3] must be multiplied by a normalizing term 9 5 Ve /~4 p « 0 a ! ,
[7]
where e is charge on an electron and «0 is the electric permittivity of vacuum. The mks field strength (%) is related to the dimensionless field’s strength by % 5 eE /~4 p « 0 a 2 ! .
[8]
Similarly, the mks total potential energy of the distribution ((3%) and mks work (0) are related to their dimensionless cousins by (3% 5 TPE e 2 /~4 p « 0 a !
[9]
0 5 We 2 /~4 p « 0 a ! .
[10]
and
All that remains in solving this problem is to specify the distribution of the charge in the dielectric. It is conventional wisdom that charge will distribute itself ‘‘uniformly’’ on the surface of a spherical conductor. ‘‘Uniformity’’ of charge distribution is a concept that cannot be applied to a discrete number of charges. For example, how does one place five charges, ‘‘uniformly’’ (i.e., so one charge is indistinguishable from the others) on the surface of a sphere? It is not possible. In fact, charges do not distribute ‘‘uniformly’’ but rather in a distribution in which the total potential energy (TPE) is a minimum. In the present study, the charge distributions were determined by first placing the charges randomly inside the dielectric droplet and then allowing them to rearrange in response the forces exerted on them until the minimum TPE distribution is attained.
FIG. 1. The field lines in the vicinity of one charge located at a radial position of 0.943 in a droplet with 100 charges and a dielectric constant of 80.
RESULTS
Using the preceding approach, it is straightforward to calculate the fields surrounding the charges in a dielectric droplet. Figure 1 shows the field lines near a charge in a droplet containing 100 charges and with an « 5 80. In many of the examples presented in this paper, the dielectric constant of 80 will be used because the « for water is about 80. The charge is located 0.057 radii from the surface. Notice the discontinuities in the field lines at the surface of the droplet. These discontinuities result from the jump boundary condition in the normal component of the potential gradient at the surface of the droplet (Eq. [2c]). Physically, these discontinuities reflect the polarization surface charge arising from the charges inside the dielectric. It is evident from the examination of Fig. 1 that there is a polarization charge induced on the surface of the droplet. A single charge in an infinite medium will be uniformly surrounded by dielectric material and will feel no net force. If that same charge were located near the inner surface of a dielectric droplet, the charge would induce a polarization charge on the surface of the droplet. This polarization charge would exert a force on the original charge, pushing it to the center of the droplet. This force will be referred to as the ‘‘polarization force.’’ A single charge in a dielectric droplet will tend to move to the center of the droplet because of this polarization force. If a second charge were added to the droplet, the charges would exert repulsive forces on one another, which would tend to push them apart, towards the surface of the droplet. This repulsive force is referred to as the interaction force because it arises from the coloumbic interaction between the charges. So, the outward interaction force is opposed by the inward polarization force. The charges locate where the interaction force balances the polarization force. The exact position of this equilibrium point depends on the value of the « and on the number of charges (N) in the droplet. If « were unity, there would be no polarization force, and the charges would be
22
M. LABOWSKY
charges, one test charge is moved, while the other N 2 1 charges are fixed in their otherwise equilibrium position. The change in the work (W) required to move the test charge from the center of the droplet to the surface is then calculated and compared with the thermal energy of the charge. Figure 3a shows the work as a function of radial distance (R) for the case of N 5 100 and « 5 80. The work is assigned a value of zero
FIG. 2. The equilibrium sphere of charge radius (Req) as a function of the number of charges (N) and the dielectric constant.
forced to the surface of the droplet by the interaction force. As « increases, the polarization force also increases. The polarization force pushes the charges away from the surface. As mentioned previously, the equilibrium position of the charges is determined by allowing the charges to move in response to the forces acting on them until the TPE is a minimum. In so doing, it was found that the charges arrange approximately on a ‘‘sphere of charge’’ within the droplet.4 The radius of this ‘‘sphere of charge’’ is less than the radius of the droplet. Figure 2 shows the radius (Req) for the equilibrium sphere of charge as a function of the number of charges (N) on the droplet. This figure has five curves, each representing a different value of «. Figure 2 clearly shows the interplay between the interaction and the polarization forces. As the number of charges increases for a fixed «, the equilibrium position moves closer to the surface of the droplet because the outward interaction force increases with increasing N. Alternatively, the equilibrium radius decreases as « increases for a given N because the inward polarization force increases with increasing «. It is noteworthy that for « larger than 10, the equilibrium position is fairly insensitive to «. The curves for all the values of « approach the limiting case of 1 for an infinite amount of charge on the droplet. Figure 2 shows the equilibrium positions of the charges on the sphere of charge. The question as to how firmly the charges are locked into that equilibrium position then arises. For example, all the charges will have associated with them a certain amount of thermal energy. How far can the charges move as a result of this thermal motion? To answer this question, one charge in the distribution will be designated as a ‘‘test’’ charge. This test charge is allowed to move in the radial direction, while all other charges are locked on the equilibrium sphere of charge. To clarify this point, if a distribution contains N 4
Except in a few cases, the charges do not locate exactly on a sphere of charge. In general, charges arrange slightly above and slightly below the sphere of charge.
FIG. 3. (a) Work as a function of radial position for a test charge in a droplet with N 5 100 and « 5 80. (b) Work as a function of radial position and « for a droplet with N 5 100. (c) Work as a function of radial position and « for a droplet with N 5 200.
DISCRETE CHARGE DISTRIBUTIONS
at the center of the droplet. As the test charge moves from the center, W is negative. This means that the charge experiences an outward interaction force which would push it from the center toward the surface. The slope of the curve is small near the center but becomes steeper (more negative) as R increases. This means that the outward interaction force acting on the charge increases with R. In the region near R 5 0.943, a potential well is reached. This potential well corresponds to the equilibrium sphere of charge point found in Fig. 2. Beyond this well, the slope of the work curve changes and becomes positive. The test charge experiences a net inward force because the inward polarization forces are larger than the competing outward interaction forces. This polarization force would become very large (infinite) if the charge were able to get infinitesimally close to the surface. The net inward force becomes stronger the closer the charge moves to the surface. Indeed, in the classical sense, the surface represents an impregnable potential barrier to the charge. Figure 3b shows the work curves for several values of « for a droplet with 100 charges. The depth of the potential well increases as the dielectric constant decreases. For small «, the outward coloumbic interaction force is more pronounced near the center of the droplet than for large «. However, even for small «, there is a strong inward polarization force when the test charge is very near the surface. The potential well is a stable equilibrium point for the charge. If the charge is perturbed from the equilibrium position, it will tend to return to that position. This is in contrast to the ‘‘Schottky Hump’’ (19, 20) located outside the droplet which is an unstable equilibrium point. Figure 3c shows the work curves for several value of « for the case of a droplet with 200 charges. The curves are qualitatively the same as those shown in Fig. 3b; however, the potential wells are deeper. The depth of the potential wells reflects the increased strengths of the interaction and polarization forces as the number of charges increases. The dependence of the depth of the potential well on N for various « is shown in Fig. 4. An examination of this figure clearly shows that as N increases, or as « decreases, the depth of the potential well increases. For droplets with small values of «, the potential wells can be very deep. Up to this point, only dimensionless quantities have been used in order to understand qualitative trends. It is now of interest to see quantitatively how tightly the charges are held to the equilibrium point. The charges will have a certain amount of thermal energy. It is important to compare this thermal energy with the depth of the potential well surrounding the charges. Consider a droplet which has a dielectric constant 80, a radius of 100 angstroms (Å), and 100 charges. In this case, the bottom of the potential well is at 0.943, or a distance of 5.7 Å from the surface of the droplet (see Fig. 3b). The depth of the potential well is 0.168 which corresponds to an energy of 3.87 3 10214 ergs (see Eq. [9]). For a droplet at 300 K, the
23
FIG. 4. The potential well depth as a function of the number of charges (N) and «.
charge would have 3kT/2 or 6.21 3 10214 ergs of thermal energy (TE). Hence, the ratio of the TE to the potential well depth is 1.6. This means that if the charge has 3kT/2 of TE, it could move to the center of the droplet and could approach the surface to a radial position of about 0.99. So, thermal motion would allow the charges to move freely about the equilibrium position. If the droplet had a radius of 50 Å, the ratio of TE to the well depth would be 0.8. The charge would be able to move only between radial positions about 0.5 and 0.985. Conversely, if the droplet had a radius of 200 Å, the TE to well depth ratio would be 3.21. The charge would be able to reach the center of the droplet and could reach a radial position greater than 0.995. It is clear from this that charges would be able to thermally move more freely in larger rather than in a smaller droplets if the larger and smaller droplets had the same amount of charge. If we looked at the case in which « were 10, the potential well is noticeably deeper. For 100 charges, the well is located at 0.947 and has a depth of 1.286 (see Fig. 3b). For a droplet with a 100-Å radius, the well locates 5.3 Å from the surface and has a depth of 2.96 3 10213 ergs. The TE to well depth ratio is 0.21, indicating that the charges are very tightly bound to the potential well. The charge would be able to move only between the radial positions of about 0.86 and 0.975, so the thermal motion is significantly restricted. The same dependence on droplet size as in the « 5 80 case would be found here. The thermal motion of the charge would be more restricted in smaller droplets than in larger ones. In the preceding calculation, only one charge was moved while the others remained locked in their equilibrium positions. The purpose was simply to show the magnitude of the potential well near each charge. If a charge in a droplet were to move in response to this disturbance, and the calculated work would be expected to be different from that calculated here. The response to the disturbance, however, will be transient and depend on the mobility of the ions in the dielectric. It is also clear from these calculations that for large « the charges will not statically locate at a specific point but will dynamically move
24
M. LABOWSKY
as a result of thermal motion. The dynamics of charge motion in dielectric droplets is a fertile area for research but one which is beyond the scope of the present paper. Up to this point, we have been concerned with the forces acting on a test charge as it moves from its equilibrium position. As has been seen, the surface polarization force acts to push the charge toward the center of the droplet. However, as the surface exerts an inward force on the charge, the charge exerts an outward force on the surface. It is important to study this force because it may play a crucial role in the ion emission process. The expression for the electrostatic pressure (3) (referred to henceforth simply as the pressure), acting on the surface of dielectric, can be found in any number of textbooks (see, for example, Ref. 22) 3 5 « 0 ~« 2 1!@% 2t 1 «% 2n #/ 2 ,
[11]
where %t and %n are, respectively, the tangential and the normal components of the electrostatic field strengths on the dielectric side of the surface. If all N charges were located at the center of the droplet, then %n 5 eN/(4p«0a2)«, %t 5 0, and the pressure would be that for a continuous charge distribution 3 cont 5 ~« 2 1!@e 2 N 2 /~4 p a 2 ! 2 «« 0 #/ 2 .
[12]
This pressure would be uniform over the entire droplet surface. 3cont serves as a useful reference pressure. It is the pressure which would be exerted on the surface of a droplet if the charge were continuously distributed on the surface of the droplet. Note that 3cont is directly proportional to N2 and has a weak dependence on « except when « is small. A dimensionless pressure (P) is defined as the actual pressure (3) divided by 3cont P 5 @E 2t 1 «E 2n #/@N 2 /« # ,
FIG. 5. The electrostatic pressure (P) exerted on the surface of a droplet in the vicinity of two adjacent charges in their equilibrium positions for a droplet with N 5 100 and « 5 80.
[13]
where Et and En are the dimensionless field strengths inside the droplet in the tangential and normal directions, respectively. This dimensionless pressure is a direct measure of the uniformity of the actual pressure. If P is unity, then the pressure is the same as if the charge were continuously distributed on the droplet. Deviations from unity indicate that the charge distribution is not continuous. In the following, the term pressure will be used to describe the dimensionless electrostatic pressure. Figure 5 shows the pressure, as a function of the fractional angular displacement (Q/Q0) between a test charge and an adjacent charge for the case of « 5 80 and N 5 100 when both charges are at their equilibrium positions (R 5 0.943). Q0 is the angle between the lines joining the adjacent charges to the center of the droplet. Q/Q0 is 0 above the test charge and 1
above the adjacent charge. On the surface above the test charge (Q/Q0 5 0), the pressure is 1.137. This means that on the surface immediately above the test charge, the pressure is 13.7% higher than that for a continuous distribution. As Q/Q0 increases, P gradually rises to a maximum (PMAX) of 1.15 at a Q/Q0 value of 0.06. P decreases beyond this maximum and reaches a minimum value of 0.99 at the midpoint (Q/Q0 5 0.5) between the two charges. This minimum is slightly below unity, indicating that the pressure is below the continuous charge distribution pressure. The pressure curve is symmetrical about the mid point and the same pressure hump is observed over the adjacent charge as was observed over the test charge for this position of the test charge. Figure 6a shows the pressure as the test charge is moved from its equilibrium position and the other charges remain locked in their equilibrium positions. As the test charge approaches the surface, the pressure noticeably increases. For R 5 0.95, PMAX is 1.22. For R 5 0.96, PMAX is 1.45. For R 5 0.97, PMAX is 2.19. So, small deviations from the equilibrium position of the test charge can result in very large deviations of the local pressure from the continuous distribution pressure. This local pressure can be several times larger than that for a continuous distribution. As was mentioned earlier, if the test charge had 3kT/2 of thermal energy, it could easily reach a radial position of R 5 0.97 even for 50-Å droplets with « 5 80. If the test charge moved to a radial position of 0.98, the maximum pressure would be more than 5 times greater than assumed using a continuous charge distribution. Note that while the pressure in the immediate vicinity of the test charge increases dramatically, this pressure disturbance decreases rapidly from the maximum. For example, for R 5 0.97 the maximum is 2.19 at Q/Q0 5 0.06, but it decreases to 1.28 at Q/Q0 5 0.15. This means that the large pressure disturbances are highly localized. The size of the highly stressed surface area can be estimated. For N 5 100, the
DISCRETE CHARGE DISTRIBUTIONS
distance between charges is of the order of 0.355 radii. A Q/Q0 of 0.15 then corresponds to a disturbance diameter of 0.106 radii or 10.6 Å for a 100-Å droplet. A Q/Q0 of 0.06 corresponds to a pressure disturbance diameter of only 0.042 radii or 4.2 Å. So, the high-pressure areas on the surface are very localized. It should also be mentioned that when the pressure above the test charge is significantly larger than the continuous distribution pressure, the pressure at the midpoint between the charges droplet is slightly below the continuous pressure. The pressure above the adjacent charge, on the other hand, is nearly the same as that as when the test charge is in the equilibrium position. Figure 6b shows the pressure curves for the case of N 5 200. It is interesting that, when the test charge is in its equilibrium position (R 5 0.96), the curve is similar to the equilibrium curve for N 5 100. As the test charge approaches the surface, however, the rise in the maximum value of P is not as great as that for N 5 100. As may be seen, PMAX is 1.37 when R 5 0.97 compared with 2.19 for the N 5 100 case. This is significant because it indicates that PMAX decreases with increasing N for
FIG. 6. The electrostatic pressure (P) exerted on the surface between a test charge and an adjacent charge as a function of the radial position of the test charge (a) for the case of a droplet with N 5 100 charges and « 5 80 and (b) for the case of N 5 200 and « 5 80.
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FIG. 7. The maximum electrostatic pressure (PMAX) exerted on the surface as a function of the number of charges and the dielectric constant for a test charge position of 0.96.
a given test charge position.5 It should also be noted that when the test charge is below its equilibrium point (R 5 0.95 curve), the value of PMAX approaches the continuous limit of unity. This would be expected because continuous distribution approximation is better the farther a charge is from the surface. The important point to be made here is that even though the pressure approaches the continuous distribution limit of unity when the charge is far from the surface, PMAX is 15% greater than predicted by a continuous charge distribution assumption when the charge is in its equilibrium position. The dependence of PMAX on N for test charge in a fixed radial position is shown in Fig. 7. This figure is a graph of PMAX as a function of N with five curves for each of the indicated values of «. This figure is for a test charge at a radial position of 0.96. All the curves show that as the number of charges increases, PMAX decreases and asymptotically approaches the continuous distribution P limit of unity. The figure is broken into two regions labeled ‘‘above’’ and ‘‘below’’ and separated by a broken line. The purpose is to indicate whether the test charge at a radial position of 0.96 is above or below its equilibrium position. For example, the equilibrium position for a distribution with 500 charges with an « of 80 is 0.975, so a test charge position of 0.96 is below the equilibrium value. PMAX can be very large for small values of N. For N 5 50 and « 5 80, PMAX is 2.5. For N 5 50 and « 5 40, PMAX is 4.0. So, for small values of N, PMAX can be several times larger than that expected by assuming a continuous charge distribution. As N increases, PMAX decreases rapidly at a rate which is inversely proportional to N2. The approach to a continuum distribution for large N evident in Fig. 7 is somewhat misleading. In this figure, the reason the Remember that PMAX is normalized by 3cont, which directly depends on N . Consequently, while the dimensionless pressure decreases with N, the magnitude of the actual pressure increases as a result of this square dependence on N. 5
2
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FIG. 8. The maximum electrostatic pressure (PMAX) exerted on the surface when all charges are in the equilibrium positions as a function of the number of charges and the dielectric constant.
continuum limit is approached is that by keeping R fixed the charges locate below their equilibrium positions when N is large. The deeper a charge is buried in the droplet, the more continuous it appears to the surface. Conversely, in Fig. 7, PMAX is large when N is small because the charges are above their equilibrium positions when N is small. So Fig. 7 shows PMAX values that tend to be too low for large N and are too high for low N. In contrast, Fig. 8 shows PMAX when the test charge is at its equilibrium position (PMAX, EQ). This figure reveals that PMAX, EQ is only a weak function N. For « 5 80, PMAX, EQ is 1.145 for N 5 50 and 1.157 for N 5 500. This means that for N 5 50 and N 5 500, PMAX, EQ is, respectively, 14.5% and 15.7% higher than would be predicted based on an assumption of a continuous charge distribution. It is significant that PMAX, EQ increases slightly when N changes from 50 to 500. The fact that PMAX, EQ is nearly independent of N means that the values of PMAX, EQ calculated here will persist and may increase slightly for extremely large values of N. It is also clear from Fig. 8 that as « decreases, the value of PMAX, EQ increases significantly. For example, for N 5 500 and « 5 40, PMAX, EQ is 1.3. For « 5 20, it is 1.7. For « 5 10 and 5, PMAX, EQ is, respectively, 2.46 and 5.1. All this indicates that the assumption of a continuous charge distribution to describe the pressure on the surface of a charged droplet is basically flawed. Even though the error associated with a continuous charge distribution may be acceptable when « is large (on the order of 15% for « of 80), the actual value of PMAX, EQ will always be larger than that predicted by a continuous charge distribution regardless of the number of charges on the dielectric droplet. DISCUSSION
As mentioned in the introduction, charge is lost by either the emission of charged daughter droplets or by the emission of ions. The present work has shown only that the surface above
charges in dielectric droplets can be locally very highly stressed. The response of the surface to that stress will be the subject of future work. It is appropriate here, however, to discuss the possible implications of the present results. Rayleigh derived an instability limit by considering the perturbation of a droplet which was initially spherical and had a continuous charge distribution. If the perturbed droplet returned to its spherical shape, it was stable, and if it did not, the droplet was unstable and would lose mass and charge. Rayleigh did not discuss the nature of the perturbation only that the perturbation existed (6). A charged droplet can be perturbed from its spherical shape in any one of a number of ways. There may be hydrodynamic forces acting on the surface. Perturbations may also result when the droplet is in a nonuniform electric field because of external field gradients, encounters with other charged droplets, or statistical fluctuations of the charge distribution. Such perturbations would disturb the spherical symmetry of the droplet and/or its charge and could cause a highly charged droplet to become unstable and lose charge. But suppose the charged droplet were in a perfectly quiescent and uniform environment so there are no external hydrodynamic or electric effects. If the droplet were considered to have a perfectly continuous charge distribution, the charge would exert a perfectly uniform pressure on the surface of the droplet. In this highly idealized situation, there would be no obvious site where an instability would occur. The importance of the present work is to show the charge distribution cannot be considered continuous. As may be seen from Fig. 8, even for very large values of N, there are nonuniform local pressures on the surface in the vicinity of the charges which can perturb the surface. In the absence of external effects, the surface immediately above any of the charges would be a potential site of instability. Further, the Raleigh instability limit which was derived assuming a continuous charge distribution may require modification in order to account for the local nonuniform pressure in the vicinity of the charges. These results allow us to make some interesting conjectures concerning ion emission from dielectric droplets. If a charge were to move infinitesimally close to the surface, it would hit an infinitely high potential wall. In a classical sense, the charge would not be able to penetrate this potential wall and would be trapped inside the droplet. In practice, however, the charge has a finite size and so cannot move infinitesimally close to the surface. Further, as the surface charge exerts a force on the charge, the charge exerts an equal but opposite force on the surface. The droplet surface will tend to deform in response to this force. The amount of deformation will depend on the ratio of the electrostatic pressure exerted on the surface by the charge to the restoring surface tension on the droplet. If this pressure ratio is small, there will be little deformation of the surface. If the pressure ratio is large, the surface will deform. As was shown earlier, the local pressure exerted on the surface can be large. This local pressure would exist even if the
DISCRETE CHARGE DISTRIBUTIONS
27
FIG. 9. The proposed mechanism for charge loss. (a) The charge approaches the surface. (b, c) The force exerted on the surface because of the charge distorts the surface and draws the charge outward. (d) The distorted surface separates from the droplet forming a charged ion. (e) If the force is slowly exerted on the surface, the surface may deform into a minicone.
charges were in their equilibrium positions but would become more pronounced if the charges moved toward the surface as a result of thermal motion. The pressure nonuniformity could be several times greater than that for a continuous distribution. This means that there can be a local stretching of the surface with the result that small bumps will tend to form in the vicinity of the charges (see Fig. 9a). These bumps may occur as a result of either a ‘‘one shot’’ encounter with a charge or a resonance induced by repeated encounters with a charge which
is oscillating about its equilibrium position. As was seen previously, the size of the bumps would be much of the size of a solvated ion. It is speculated that these bumps enlarge (Fig. 9b). As they enlarge, the surface moves away from the charge, allowing the charge to be drawn toward the surface. At this point, one of two possibilities emerge. Either the bump expands rapidly and detaches from the surface (Fig. 9c, d), encapsulating the charge and liberating a (solvated) ion, or the bump expands slowly and stretches the surface to form a
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M. LABOWSKY
microscopic cone, pulling in several neighboring charges (Fig. 9e). Roellgen (23, 24) has suggested that ion evaporation is not likely to occur and is critical of the work of Iribane and Thomson (I&T) (19) claiming that solvated ions are emitted when a Raleigh instability somehow degenerates into a series of microscopic cones. Roellgen does not, however, propose a mechanism for this process. The present work may point to such a mechanism, although confirmation must await future calculations. The present work focused only on the electrostatic fields near discrete charges. Several physical properties will also play a role in the ion emission process. One such property is the size of the ion. High electrostatic pressures are found when a charge is very close to the surface. A large ion may not physically be able to get sufficiently close to exert these large pressures on the surface. This and other considerations will be the subject of future work. This future work may follow along the lines of the calculations of Basaran (11–15). Basaran has provided detailed calculations of the large-scale deformation of liquid droplets and liquid in tubes in response to electrical fields. Basaran used continuous charge distributions and boundary integral methods in his work. The future work will attempt to incorporate discrete charge distributions into the boundary integral methods in an attempt to see how the droplet surface responds to the localized electrostatic pressure near each charge. When this future work is completed, comparison of the theory with experiments will be possible. CONCLUSIONS
This paper presents a method for calculating the electrostatic properties of dielectric droplets with discreet charge distributions. Several interesting observations can be made from the results. Charges in a dielectric droplet will tend to locate slightly below the surface of the droplet in a position in which the inwardly directed polarization force equals the outwardly repulsive force of the adjacent charges. The electrostatic pressure exerted on the surface of the droplet is highly localized above a charge. This pressure can be extremely high if the charge strays toward the surface from its equilibrium position. The fact that the local pressure can be very high and the charges are located slightly below the surface leads to the conjecture of an ion emission mechanism in which an ion pierces the surface of the droplet. Further, the fact that the electrostatic pressure near a charge will always be higher than that calculated using a continuous charge distribution means that the classical Rayleigh instability limit, which is based on a continuous charge distribution, may require modification. Clearly more work needs to be done in this area. The present work, in fact, has raised more questions than it has answered.
However, the questions that have been raised are important and deserving of further study. The next phase of this research would be to examine the dynamics of the charge interactions and to model the local stretching of the surface by the charges with the intent of calculating the rates at which singly charged ions are emitted from the droplets. It would also be important to see if local stretching is a sufficient disturbance to initiate a large-scale instability for droplets near the Rayleigh limit. In conjunction with this next phase of research, it would be important to study the dynamics of charges which are bound to neighboring charges with the hope of better understanding the multiply charged ion emission process and to see if and when there is preferential ion emission for multiply charged ions. ACKNOWLEDGMENTS The author thanks Prof. J. B. Fenn of Virginia Commonwealth University and Prof. Juan Fernandez de la Mora of Yale University for their many thoughtful discussions, insights into this problem, and help in preparing this manuscript. The author also thanks Anna and Sarah Labowsky for their help in preparing the graphs.
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