© Jiří G.K. Ševčík, 2002

The random process of a separation of an unknown mixture is characterized by its ultimate uncertainty UU. For one-dimensional system it holds that

Eq. 1 comprises two terms, the number of analytes Z_n , and the efficiency of the separation system. By solving the equation for separation efficiency we obtain

where Z_n is a number of analytes, r_ij is relative retention for analytes i to j, and R_ij is their resolution.

Because it holds that

the equation for the ultimate uncertainty can be rewritten in the form

For number of analytes Z_n separated from an unknown mixture it can be assume that the analytes will be randomly distributed along the separation axis defined in terms of the retention time, retention volume, retention index, etc. Thus the probability of the retention position x is expressed by the normalized function

The probability that two peaks differ by a minimal distance x₀ is given by Eq. 8. It holds that

where n_c is called the peak capacity (the maximum number of separated peaks within distance x₀ ) and can be expressed for different chromatographic modes. For the isothermal mode holds that

where R is the resolution required and the peak width is expressed as (a+bt_x ).

For a linear temperature program mode (assuming a constant peak width) it holds that

When comparing Eqs. 10 and 11 it can be seen that the temperature program mode is superior to the isothermal mode and decreases the ultimate uncertainty of measurements. Fig. 1 shows the effect of the efficiency only (a factor 3.9 ) while the effect of polarity (due to the temperature change of the stationary phase) is not included.

Various number of components, n, can be found within separation distance x₀ . For the probability of overlapping n components holds that

The total number of overlapping clusters with the same number of peaks is

where the total number of peaks presented in the separation space X is given by the sum of p_n . It holds that

Figure 2 shows the number of singlets in one-dimensional mode as the function of the number of plates for m =100 and k <1;10> .

It is necessary to differentiate between multidimensional chromatopgraphic systems and multidimensional switching systems.

A chromatographic dimension is characterized by a constant value of the distribution constant of an analyte within its analysis. Thus experimental arrangements leading to its change within one run (such as different stationary phases, different temperatures) belong to multidimensional chromatography systems.

A switching dimension is characterized by a straight “sample inlet-separation part-detector” within one analysis run.An experimental arrangement leading to multiplication of any part of the trajectory belongs to multi-dimensional switching systems.

It is evident that there can exist multidimensional chromatography without mulditimensional switchning (temperature program modes) and multidimensional switching without multidimensional chromatography (pre- or post-column splitting).

Hyphenated techniques can be both multidimensional separation systems (HPLC-GC) and multidimesional switching systems (FID-MS). Interfaces of different techniques (GC-FTIR) are very often considered as hyphenation labeled but are not necessarily multidimensional.

In multidimensional chromatography, the distribution constant is diferent in each part, and thus the analytes will behave different by them. Therefore, the separation in a one-dimensional system will be enhanced in proportion to the number of chromatographic dimensiones d. Thus an equivalent of Eq. 13 holds for d-dimensional space

Equation 16 demonstrates the improvement attained by a multidimensional separation. The peak capacity increases with the number of dimensiones and in an ideal case (non intercorrelated dimensions) it corresponds to the product of the single dimension capacities. It holds that

The dependence of a relative number of peaks (p/¯m) on a total number of components (¯m) for 1 to 4 dimensional chromatographic systems is shown in Fig. 3 (for k<0;10> and R=1). Under ideal conditions, the number of components increases 31 times with each dimension or the probability of identification improves 31 times for the same number of components.

It has been shown that an added dimension reduces ultimate uncertainty. What are the criteria for an additional dimension?

The most general crirerion for applicability of a new dimension is evaluation of the information content. The information theory is based on evaluation of the uncertainty (entropy) prior to an experiment and after it. The uncertainty is thus related to probability that some input and output events will happen. As the input we can use the analyte identity for i = 1 to n, expressed as P(I_i), while an output corresponds to the probability that there will be a measurable signal (larger than LOD) P(O_k). The output-input relation is probability P(O_k/I_i) which corresponds to the probability that the signal O_k will be measured only when there will be input I_i and can be expressed by Bayes equation (conditional probability). It holds that

Probability P(I_i/O_k) espresses situation that there will be input I_i (component i will be present) when there will be measured output O_k . The relation of all the probabilities is shown in Fig. 4 and it holds for all of them that they are positive, normalized and additive.

For entropy (“uncertainty”) of the above mentioned statuses it holds according to Shannon that:

Information (in terms of the information content) is defined as the diference between the uncertainties before and after analysis and it holds that

where I(d) expresses intercorrelated informations.

To express the information content of multidimensional systems according to Eq. 23, following approches are used:

• integration approach,
• approach base on assumption of the normal distribution of the data and of the error of measurement,
• a histogram approach,
• an approach based on maximum peak capacity,
• an approach based on real peak capacity.

Approaches based on the peak capacity are best suited for espression of the information content of multidimensional systems then they reflect both the separation system properties (efficiency, selectivity, switching, etc.) and the sample behaviour (peak overlapping). For information content holds that

Details on the information content calculation are beyond the scope of this lecture. As an example, the expressions and values for a multidimensional system with columns in series are given in Table 1.

Validity of Eq. 23 constitutes the background for purposeful and substantiated development of analytical instrumentation. It comprises every step of analysis which leads to an improvement in an a posteriori probability, different injection techniques, stationary phases, columns and their combinations, detectors, etc.

Generally there are three approaches for development of a new sophisticated instrumentation:

• an application of an combinatorial approach for improvement of conditional probabilities,

• an application of hyphenation for improvement of the analyte signal probability,

• an application of adaptive expert systems for minimalization of model residual error.

Multidimensional separation hardware systems are serial, parallel or combined. In a serial system, the sample separation is carried out first under given conditions characterized by distribution constant value K_d1 and then under changed conditions represented by a changed distribution constant K_d2 (K_d1 ≠ K_d2 ). This general condition of multidimensional separation holds for parallel systems as well but the sample is split into different columns and analysed simultaneously. Schematics of the two separation systems are given in Fig. 5.

There are numerous column combinations and switching modes applied in multidimensional chromatography. Switching modes are very often used to improve efficiency of the analysis in terms of information flow and information efficiency (derived from the information content). In Table 2 are listed most frequently used modes in the multidimensional chromatography.

As shown above, much greater data volumes are produced in multidimensional chromatography compared to one-dimensional chromatography. Hence the working with multidimensional separation system data handling systems is more complex, and very often linked to a chain

• peak identification (processes of a noise reduction, integration, deconvolution),
• peaks correlation (similarity linked to similar analyte groups),
• analyte identification (match with standards).

It is evident that both hardware (mainly miniaturization) and software (mainly deconvolution routines and methods of multidimensional statistics) components are of essential importance for multidimensional chromatography development.