Capillary electrophoresis is renowned for its high separation efficiency, but several physicochemical and operational factors influence this performance. This lesson explores the roles of electrolytic processes, Joule heating, and field-driven ion migration in shaping the efficiency and resolution of CE separations. We will also examines why CE uniquely adheres to exclusively the molecular diffusion term of the van Deemter equation, and introduce concepts such as electromigration dispersion. These insights are essential to understanding the true potential and practical limitations of CE in analytical separation science.
Learning Goals #
After this lesson you should be able to
Describe the electrolytic processes occurring at the electrodes during a CE run.
Explain how changes in pH and ion concentrations in the buffer reservoirs can affect separation quality.
Evaluate the efficiency of a CE separation using theoretical plate numbers.
Understand why only the molecular diffusion term (B-term) applies in the van Deemter framework for CE.
Describe the causes and consequences of Joule heating, and explain strategies for thermal management.
Define resolution in CE and identify the factors that influence it.
READ SECTION 5.2.1
General Instrumentation
1. Electrochemical processes #
Charge transfer occurs in electrophoretic systems. While ions move through the capillary, electrons flow through the power supply, the electrodes and the connecting wires (see Figure 1).
Figure 1. Schematic illustration of the principle of electrophoresis. In this figure, electrophoresis occurs inside a capillary, but it can also be conducted in other formats.
As a result, electrolytic process occur at the electrodes. At the positive electrode oxidation occurs, usually of water under the formation of H+ (H3O+) ions:
Equation 5.21: 2\text{H}_2 \text{O} \rightarrow \text{O}_2 (\text{g}) + 4 \text{H}^+ + 4e^-
At the negative electrode reduction takes place, also usually of water under the formation of OH– ions:
Equation 5.22: 4\text{H}_2 \text{O} + 4e^- \rightarrow 2\text{H}_2 (\text{g}) + 4 \text{OH}^-
The electrons produced at the positive electrode flow through the power supply to the negative electrode.
The number of moles of \text{H}^+ ions formed at the positive electrode (\Delta n_{\text{H}^+}) follows from Faraday’s law, i.e.
Equation 5.23: \Delta n_{\text{H}^+}=\frac{Q}{F}=\frac{I \cdot t}{F}
EXERCISE 1
A CE experiment runs during 20 min and the measured electrical current is 20 µA. What is the amount of H+ ions formed in the buffer vial that contains the positive electrode? Calculate the amount in μmol and round to two decimals. Use Equation 5.23 and note that 1 A = 1 C\cdots-1.
This is correct!
Unfortunately this is not correct. Assume that Faraday’s constant is 96485 C\cdotmol-1. Did you check the rest of the units? Is the equation unit neutral?
Because the BGE reservoir at the positive electrode gets more acidic during the run, while that at the negative electrode gets more basic, the pH in the reservoirs may shift and the pH may start to vary along the separation capillary. To minimize these effects, BGE should have significant buffering capacity, separate vials should be used for flushing of the capillary, and the inlet and outlet reservoirs should be replaced by fresh ones frequently.
READ SECTION 5.1.3
Band Broadening in CE
2. Efficiency #
We will now investigate the efficiency of capillary electrophoresis. Let’s do some calculations. To aid us, we’ll base ourselves on the electropherogram in Figure 2.
Figure 2. Separation of peptides by capillary electrophoresis. 1: gonadorelin, 2: des-tyr-met-enkephalin, 3: met-enkephalin, 4: [D-ala,D-met]-met-enkephalin. For peak 2, W_{0.5}=0.78, and for peak 3 W_{0.5}=0.92. Data courtesy of Prof. Govert Somsen (Vrije Universiteit Amsterdam).
EXERCISE 2
Estimate the number of theoretical plates (N) for the two peaks shown in the electropherogram of Figure 2. The width at half height (W_{0.5}) are provided in the caption of the figure. To help you, the migration time of peak 2 is 264 seconds and that of peak 3 is 295 seconds. Equation 1.26 from Lesson 1 is valid. Also remember that W_{0.5}=2.35 \sigma.
Enter the plate number of the first peak as an integer (no decimals). The answer for the second peak will be provided in the answer feedback.
This is correct! For peak 3 it is 569611! These are VERY high numbers compared to LC!
This is not correct. Try again! The answer should be above 500000! Make sure you keep the units consistent.
Clearly, capillary electrophoresis can be called an efficient separation technique. Why is this the case?
EXERCISE 3
When considering the traditional van Deemter equation,
which effect(s) do you expect to play a role in capillary electrophoresis?
This is correct! There are no particles in the capillary, so the A term disappears (as is the case for open-tubular chromatography columns; see Section 1.7.6). In CE there is no pressure-driven flow, so no parabolic flow profile. The flow due to EOF has essentially a plug-flow profile (i.e. flow velocity independent of the position in the capillary), except from the narrow region close to the wall (the electrical double layer). Thus, the C_{\text{m}} term disappears. Finally, there is no stationary phase, so the C_{\text{s}} term also disappears.
There are no particles in the capillary, so the A term disappears (as is the case for open-tubular chromatography columns; see Section 1.7.6). In CE there is no pressure-driven flow, so no parabolic flow profile. The flow due to EOF has essentially a plug-flow profile (i.e. flow velocity independent of the position in the capillary), except from the narrow region close to the wall (the electrical double layer). Thus, the C_{\text{m}} term disappears. Finally, there is no stationary phase, so the C_{\text{s}} term also disappears.
Only the B term is expected to play a role in CE, which explains the high plate numbers that can be achieved, especially for high-molecular-weight analytes, such as proteins. However, there are some caveats, these mainly arise from injection and detection band broadening, which we will consider in the next lesson, and from Joule heating, which we will consider next. If we assume all processes to be independent, we can write for the total band broadening in CE
To achieve a very high efficiency in CE it is imperative that the other contributions are small relative to that of molecular diffusion, i.e.
We can quantitatively describe the molecular diffusion using the Einstein equation
Equation 5.12: \sigma^2_{\text{z}}=2D_{\text{m}} \cdot t_{\text{e}}
where \sigma^2_{\text{z}} is the variance of the peak in length units and t_{\text{e}} is the time at which the peak emerges at the detector, which is determined by the effective length of the capillary (L_{\text{eff}}; from the inlet to the detector), the apparent electrophoretic mobility and the field strength
Equation 5.14: t_{\text{e}}=\frac{L_{\text{eff}}}{\mu_{\text{app},i} \cdot E}=\frac{L_{\text{eff}} \cdot L_{\text{total}}}{\mu_{\text{app},i} \cdot V}
where L_{\text{total}} is the total length of the capillary (between the positive and negative electrodes). If molecular diffusion is the only source of band broadening we obtain
Equation 5.15: N=\frac{L_{\text{eff}}}{\sigma^2_{\text{z}}}=\frac{L^2_{\text{eff}}}{2D_{\text{m}} \cdot t_{\text{e}}}=\frac{\mu_{\text{app},i} \cdot V}{2D_{\text{m}}} \cdot \frac{L^2_{\text{eff}}}{L_{\text{eff}} \cdot L_{\text{total}}} \approx \frac{\mu_{\text{app},i} \cdot V}{2D{\text{m}}}
where the last approximation is valid if the detection window is close to the end of the capillary. Thus, in ideal CE a high voltage results in a high efficiency, irrespective of the length and the diameter of the capillary and proportional to the voltage applied between the electrodes.
EXERCISE 4
In ideal CE, how many plates can be expected for a small analyte anion with \mu_i = 50∙10-9 m2∙V-1∙s-1 and D_{\text{m}}=10-9 m2∙s-1, when 20 kV is the applied voltage and \mu_{\text{EOF}}= 100∙10-9 m2∙V-1∙s-1? Report an integer (no decimals).
This is correct! For an anion the electrophoretic mobility is in the direction opposite to the EOF.
This is incorrect! Note that for an anion the electrophoretic mobility is in the direction opposite to the EOF, so that \mu_{\text{app},i}=\mu_{\text{EOF}} - \mu_i.
3. Heat dissipation #
We already learned in class CE1 that an electric current runs through the conductive BGE and that this causes heat to be developed (Q=P \cdot t=V \cdot I c\dot t). To minimize heat production and the effects of the resulting convective flows, capillary electrophoresis was developed. However, although heat effects are much smaller in CE, they cannot be completely avoided. Heat is developed throughout the capillary and dissipated only through the wall. Heat transport through the wall implies that radial temperature gradients will develop, while the presence of an EOF leads to axial temperature gradients. Electrophoretic mobility varies with temperature, for example through changes in the viscosity of the solution, and radial variations in velocity lead to band broadening.
For the electrophoretic current we may write
Equation 5.17: I = \frac{\pi}{4}d^2_{\text{c}} \cdot \kappa \cdot E
where d_{\text{c}} is the inner diameter of the capillary (in m), E the field strength (in V∙m-1), and \kappa the conductivity of the BGE (in \Omega-1∙m-1), which is given by
with the sum taken over all ions present in the BGE with charge z_i, ion mobility \mu_i (in m2∙V-1∙s-1), and concentration c_i (in mol∙m-3 or mmol∙L-1).
EXERCISE 5
What is the conductivity of a 10 mM NaCl solution? (\mu_{\text{Na}^+}[/latex] = 52∙10-9 m2∙V-1∙s-1; \mu_{\text{Cl}^-} = 79∙10-9 m2∙V-1∙s-1). Report your answer with three decimals in \Omega-1∙m-1.
This is correct!
This is incorrect. Try \kappa=96485 \cdot (1 \cdot 52 \cdot 10^{-9} \cdot 10 + 1 \cdot 79 \cdot 10^{-9} \cdot 10).
The power produced in the capillary follows from P=I \cdot V (in W) or the power per m of capillary length from P'=I \cdot E (in W∙m-1). From here it is difficult to calculate the increase in temperature.
In an adiabatic system (no heat dissipation through the wall) this is determined by the heat capacity of the BGE. In a real system, also the heat transfer from the BGE to the capillary wall, the thickness and thickness and heat conductivity of the fused silica, the heat transfer from the fused silica to the coating on the outside of the capillary, the thickness and thickness and heat conductivity of the coating, the heat transfer from the coating to the surroundings, and the temperature and convection of the surroundings play a role.
Empirically, this is all collected in a heat-transfer resistance r_{\text{ht}} (in K∙m∙W-1) and
is assumed. To minimize the heat produced, the current should be kept low. A high voltage and field strength are desirable to achieve a high efficiency. To keep the current low, a BGE with a low conductivity should be used (typically less than 1 Ω-1m-1)). Heat dissipation through the wall helps minimize heating of the capillary and the formation of an axial temperature gradient (in the direction of the EOF), but causes a radial temperature gradient, which in turn causes band broadening. The best answer to this dilemma is to use a narrow capillary (small inner diameter, d_{\text{c}}).
4. Resolution #
Resolution is defined in the same way as in chromatography and, when band broadening is dominated by diffusion, we find
where \bar{\mu_{j,i}} is the average electrophoretic mobility of the two analytes considered and \bar{\mu_{\text{app},j,i}} is their average apparent electrophoretic mobility. This equation shows that high resolution can be obtained at high voltages (resulting in high efficiencies), for high-molecular-weight analytes (lows diffusion coefficients), and at when they migrate in the direction opposite to the EOF (low apparent mobility).
Eva Smolková-Keulemansová (1927-2024, Czech Republic)
Eva Smolková-Keulemansová was incarnated in the Nazi concentration camps in WWII as a teenager. She managed to survive, but her parents did not. She returned to study in Prague and worked as a professor at Charles University for many years. She had a long and productive career in analytical separation science and she was a leading advocate for the field. Eva started her research in gas chromatography, but also contributed to liquid chromatography and , especially, electromigration techniques. Her use of cyclodextrin for chiral separations with capillary electrophoresis and isotachophoresis was especially eye-catching.
5. Electromigration dispersion #
The electrical current is constant along the capillary and because the electric field strength and, therefore, the migration velocity of analytes will vary. If the sample zone has a lower conductivity than the BGE, the field strength in the zone will be higher and analytes will focus at the front of the sample zone. This effect may be used for large-volume injection of dilute (low conductivity) samples. This is known as sample stacking. When the conductivity of the sample is higher than that of the BGE, the opposite will occur and bands will be broadened. Remedies include reducing the sample size, dilution of the sample, and increasing the concentrations of the ions in the BGE.
Another problem arises if the conductivity of the analyte ions is very different from that of the ions in the BGE. This causes the local conductivity – and thus the field strength – to be different at the position of an analyte band. Moreover, the field strength will depend on the analyte concentration, i.e. it will vary across the peak. As a consequence, the migration velocity of the analyte () will vary. This will lead to band broadening and in extreme cases to characteristic “triangular peaks”. The effect is known as electromigration dispersion.
Figure 4. If the mobility of the analyte ion is higher than that of the BGE ion, fronting results in triangular peaks as shown.
If the mobility of the analyte ion is lower than that of the BGE ion, the top of the peak moves faster than the flanks, resulting in tailing peaks. If the mobility of the analyte ion is higher, fronting peaks results or triangles as shown in the figure. Because the effects of electromigration dispersion are greater at high analyte concentrations, it is also referred to as overloading.
Remedies include reducing the sample size (if detection sensitivity permits) or, more fundamentally, choosing BGE ions with mobilities similar to that of the analytes.
Concluding remarks #
Capillary electrophoresis achieves outstanding separation efficiencies due to its plug-like flow profile and the absence of a stationary phase. This lesson emphasized the unique physical properties that distinguish CE from chromatographic techniques, particularly its dependence on molecular diffusion as the dominant source of band broadening. However, achieving optimal performance requires careful control of variables such as voltage, current, buffer composition, and sample conductivity. By understanding and managing factors such as heat dissipation and electromigration dispersion, practitioners can take full advantage of CE’s capabilities in resolving complex ionic mixtures.
