Minimal ReactionâDiffusion Model of Micromixing during Stopped

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A: Kinetics, Dynamics, Photochemistry, and Excited States

A Minimal Reaction-Diffusion Model of Micromixing during Stopped-Flow Experiments Tamás Ditrói, and Gabor Lente J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b02879 • Publication Date (Web): 06 Jun 2018 Downloaded from http://pubs.acs.org on June 6, 2018

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The Journal of Physical Chemistry

A Minimal Reaction-Diffusion Model of Micromixing during Stopped-Flow Experiments

Tamás Ditróia, Gábor Lenteb*

a

National Institute of Oncology, Department of Molecular Immunology and Toxicology,

Budapest, Hungary b

University of Pécs, Department of General and Physical Chemistry, e-mail:

[email protected]

Abstract The reaction-diffusion equation was used to simulate kinetic curves measured in a stoppedflow instrument in order to understand the origin of micromixing effects. The partial differential equations were solved both numerically and by a more analytical approach using Fourier series. A fully analytical solution was obtained for the diffusion only case (when no reaction occurs). Comparisons with the results of numerical calculations showed that very reasonable analytical approximations were obtained for the diffusion-reaction case. The simulations could readily reproduce the saturation of the pseudo-first order rate constants with an increase in the concentration of excess reagent, a phenomenon first observed about 30 years ago. From the results, it can be concluded them further improvement of the performance of stopped-flow instruments is not possible by simply reducing the dead time, the efficiency of the mixing is the primary limiting factor.

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Introduction Studying the kinetics of fast reactions presents several serious technical problems. There are a number of techniques that can initiate a chemical reaction without mixing, by a very fast, highly intense change of certain external conditions (e.g. temperature jump, pressure jump, flash photolysis). As a rule, the jump techniques are only suitable for investigating reactions close to equilibrium, whereas the use of flash photolysis requires that one of the reagents should be possible to generate by a short high-intensity flash of light. These are severe requirements that strictly limit the range of applications of these methods.1-3 For most practically irreversible reactions, the only realistic option to start the chemical process is mixing two reagents. The time scale of mixing must be shorter than the time scale of the reaction itself in order to obtain useful information. In gas phase processes, this is often a challenging problem, but one that can be solved through some resourcefulness. For fast reactions in the solution phase, on the other hand, the problem of mixing is a fundamental one. Manual mixing typically takes 1-10 s. As early as in 1923, automated mixing was invented in a continuous flow instrument and successfully used for studying the reaction of hemoglobin with carbon monoxide.4-5 The mixing time in these studies was about 1 ms, which is an improvement of roughly three orders of magnitude over the manual alternative. Yet the large volumes of solutions required for the experimental studies with continuous flow made this technique prohibitively expensive for most practical applications. To solve this problem, the stopped-flow method was invented in 1940.6-7 The essence of the technique is a specially designed mixer that makes highly turbulent flow and therefore fast mixing possible. As the mixed solution only flows while it conveniently fills the observation cell, the need for large amounts of reagents was eliminated. Instead, detection devices with very high time resolution were necessary because the chemical process was followed in real


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time instead of the stationary state approach of continuous flow experiments. The mixing time was not improved significantly, but subsequent advancement in electronics made the stoppedflow method common equipment in kinetic laboratories all over the world by the 1970s. This fact inspired kineticists to seek a fundamental understanding of the physical processes that limit the performance of this method.8-17 The design of a typical stopped flow instrument is such that the time performance limits are characterized by two constants.11-12 The first is called the dead time (td), which is most commonly defined as the time necessary for the solution to reach the observation cell from the point of mixing. The second one is less regularly mentioned in the literature but is equally important: it is the filling time (tf), which means the time necessary for the solution to proceed from the inlet point of the observation cell to the outlet point.17 Both parameters can be determined from the physical performance parameters of an instrument (linear and volumetric flow rates, mixing chamber and observation chamber volumes), but it is more typical to determine them in a calibration process using test reactions.11-12,17 The dead time causes some initial part of the reaction to proceed undetected and is typically estimated from the region of the detected kinetic curves while the solution still flows (so in fact using the principles of continuous flow). The filling time causes inhomogeneity in the observation cell and can be determined from a more complex analysis of detected test kinetic traces.17 Another source of fundamental time limitation was uncovered by the research group of Dale Margerum in the mid-1980s.13 Using the kinetic method of flooding, they noticed that the experimentally measured pseudo-first order rate constants above 100 s-1 were significantly lower than expected whenever extrapolated values or reference data from another method were available. They interpreted this finding by assuming the physical process of micromixing: even if two solutions seem macroscopically fully mixed in the instrument, it



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takes some additional time before all concentrations become uniform. They introduced a purely empirical formula to describe this effect in a quantitative manner:13 1 k obs

=

1 k mix

+

1 kΨ

(1)

In this formula, kobs is the detected pseudo-first order rate constant in a stopped-flow experiment, kmix is the characteristic mixing rate constant of the instrument (independent of the process studied) and kΨ is the actual pseudo-first order rate constant of the reaction. It can be seen that for high values of kΨ, their contribution to kobs will be small, which has the physical explanation that it is micromixing that primarily governs the observed signal rather than the faster chemical process. With the introduction of Eq. 1 and subsequent progress in technology,13-17 it has become possible to measure kΨ values up to about 1500 s−1 in wellmaintained stopped-flow instruments. However, the concept of micromixing and the use of Eq. 1 had to be retained even when further attempts were made to achieve faster mixing in the pulsed accelerated flow technique.18-24 The performance of pulsed accelerated flow was only marginally better than that of stopped-flow, so the former technique did not become popular. It appears that recent developments in the stopped flow technique were primarily in the field of data processing. The design of the mixer and its performance is the same as in the 1970s. The rate constant correction shown in Eq. 1 is still used in some more recent studies,2526

but it is more typically to avoid the range of pseudo-first order rate constant above 100 s−1.

This leaves more than an order of magnitude of the theoretically accessible time scales unused for most purposes. The typical dead time of a stopped-flow instrument is still about 1 ms,10,17 which means only the last 22% can be detected for a pseudo-first order reaction with kΨ = 1500 s−1. Yet the mathematics of the pseudo-first order approach is highly forgiving in this regard: if the signal is sufficiently large (e.g. because of a high molar absorption coefficient), in principle, even measuring the last 1% of the conversion is sufficient to determine a pseudo-


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first order rate constants reliably. So the value of the dead time alone would not be preclude measuring kΨ values up to 5000 s−1, and the time resolution of absorbance measurement is also sufficient to achieve this. Given these considerations, it is clear that the phenomenon of micromixing deserves more detailed theoretical studies. Furthermore, since the turn of the millennium, there has been rapid improvement in the field of microfluidics including miniaturized mixing devices.27-34 The most important physical parameter that determines the dead time of a stopped-flow instrument is the dead volume of the mixer, which is typically about 20 mm3. For modern microfluidic technology, this is already a considerably large volume and it is conceivable that much smaller mixers could be manufactured without a loss of performance. However, such the practical use of such a device necessitates much better understanding of micromixing effects in stopped-flow experiments. The main objective of this work was to set up a minimal theoretical model that interprets the purely empirical formula13 given in Eq. 1. As the reaction-diffusion equation proved to be useful in a recent work to handle similar inhomogeneity problems in laser flash photolysis experiments,35 we sought to describe micromixing with this theoretical methodology.

Experimental Section The primary goal of this work was theoretical modeling, yet the phenomenon to be interpreted was reproduced experimentally in the system routinely used for calibrating stopped-flow instruments. The sample kinetic measurements were performed with an Applied Photophysics SX-18 MV Stopped Flow Reaction Analyzer using 10 mm optical path length. Chemicals used in this study were of analytical reagent grade and purchased from commercial suppliers: dichloro-indophenol (Aldrich), ascorbic acid (Reanal), hydrochloric acid (Reanal), sodium



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chloride (Reanal), 1-propanol (Sigma). All solutions were prepared with doubly deionized and ultrafiltered water obtained from MILLI-Q RG (Millipore) water purification system.

Results and Discussion Experimental data. In order to demonstrate the phenomenon that we aimed to interpret, a routine calibration of the stopped flow instrument was carried out based on the reaction between dichloro-indophenol and ascorbic acid in an acidic medium.10 This process is known to be strictly first order with respect to both of its reagents at constant pH. If experiments are designed under flooding conditions with ascorbic acid in excess, pseudo-first order curves can be measured with kΨ values that are directly proportional to the concentration of ascorbic acid:

k Ψ = k r [Asc]

(2)

The measured pseudo-first order curves were fitted to exponential functions to determine the experimental kobs values.10 The dependence of these kobs values are shown in Figure 1. This graph does not show the expected direct proportionality as the chemical reaction is coupled with micromixing as described in Eq. 1.13 Combining this with Eq. 2, a formula can be derived that gives the concentration dependence of kobs:

k obs =

k mix k r [Asc] k r [Asc] + k mix

(3)

This formula was fitted to the experimental data. A reasonably good agreement was found (as shown by the solid line in Figure 1) and the optimized parameter values were determined as kmix = (4.2 ± 0.2) × 103 s−1 and kr = (5.4 ± 0.2) × 104 M−1 s−1.


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Figure 1 Pseudo-first order rate constants measured in the reaction between dichloroindophenol and ascorbic acid in an acidic medium as a function of ascorbic acid concentration (excess reagent). The solid curve reflects the best fit to Eq. 3, whereas the dotted straight line shows the expectation in the absence of micromixing.

This routine procedure was also used to determine the two time constants that characterize the performance of the stopped flow instrument, the dead time10 and filling time17: td = 1.12 ± 0.13 ms and tf = 0.95 ± 0.30 ms. These values are not significantly different from each other, which is in agreement with the fact that the dead volume and the cell volume are identical (20 mm3) in the flow line. A minimal model of micromixing. Figure 2 shows two different views of the flow line of the stopped flow instrument. Two reactants, A and B are mixed in a jet mixer, and the solution passes through a small dead volume until it reaches the quartz cell, which has dimensions of 10 × 2 × 1 mm. The detection is in the direction of the flow when 10 mm optical path length is used. An option of this instrument is to use 2 mm optical path length. In this case, the detection is perpendicular to the flow, but the beam still extends to almost the whole 10 mm length of the cell. 7 ACS Paragon Plus Environment


Figure 2 Two different views of the flow line of a stopped-flow instrument.

The main objective of this study was to develop a minimal model that can already interpret micromixing effects and gives at least a semi-quantitative interpretation of the empirical correction method shown in Eq. 1. To obtain a model of mixing, spatial inhomogeneties must be described at least in one dimension. Fortunately, the physical shape of the flow line in this case offers an obvious one-dimensional mixing model shown in Figure 3, where it is assumed that as a result of macromixing, very thin solution layers (with thickness of d) of the two reagents are formed, whose boundaries are parallel with the direction of the flow. Only one of the reagents A and B are present in each layer in an alternative fashion. Micromixing between these thin layers occurs only through diffusion. It should be noted that some inhomogeneity will also form along the 10 mm length of cell in the direction of the flow. However, our earlier study on flash photolysis modelling35 clearly showed that at least an hour is needed for diffusion to have a noticeable effect at this length range, so it can be safely neglected in a stopped-flow reaction with a half-life of milliseconds.


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Figure 3 One-dimensional model of micromixing within the observation cell of the stoppedflow instrument. The thickness of the layers is very small (about 2-6 µm) compared to the length of the cell (1 cm). In the forthcoming sections, a diffusion-reaction approach will be presented to describe the concentrations of the reagents. The concentration of A as a function of time and the single spatial coordinate will be denoted by A(t,x). Similarly, B(t,x) will denote the concentration of B. Finding suitable functions A and B and averaging them in the cell provides an estimate of the macroscopically detectable concentrations. The diffusion-reaction equation. The usual diffusion-reaction equation36 will be used here to define A(t,x) and B(t,x). The one-dimensional form of this equation is given as:

∂A(t , x) ∂ 2 A(t , x) = DA − kA(t , x) B(t , x) ∂t ∂x 2 (4) 2

∂B(t , x) ∂ B (t , x) = DB − kA(t , x) B (t , x) ∂t ∂x 2 As Figure 3 shows, there are many thin solution layers of identical length in the postulated model. In fact, in order to find an analytical solution of Eq. 4, it is quite practical to assume that the number of such layers is infinitely large. In this way, a fully periodic boundary condition can be written for any time instant: 9 ACS Paragon Plus Environment


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A(t , x) = A(t , x + 2d ) (5)

B(t , x) = B(t , x + 2d ) The initial conditions are then given by:

2 A if 0 ≤ x < d A(0 , x) =  0  0 if d ≤ x < 2d (6)

 0 if 0 ≤ x < d B (0 , x ) =  2 B0 if d ≤ x < 2d Numerical integration with standard algorithms used for partial differential equations can be used to solve Eq. 4. However, we will also provide a sequence of thought leading to easy-to-use analytical solutions. As the problem is fully periodic in space, a method based on Fourier series can be used advantageously here.37 So functions A(t,x) and B(t,x) are given in the following form with coefficients aj(t) and bj(t) that depend on time but not on the spatial coordinate:

A(t , x ) =

∞

∑ a j (t )e jixπ / d

j = −∞

(7)

B(t , x ) =

∞

∑ b j (t )e jixπ / d

j = −∞

As A(t,x) and B(t,x) are real functions, Re aj(t) = Re a−j(t) and Im aj(t) = − Im a−j(t) are true for any value of t and similar equations are valid for coefficients bj (Re means the real part of a complex number, whereas Im is the imaginary part). The known Fourier series of the square wave function allows the initial conditions to be rewritten into the following form:37


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a 0 (0) = A0 a 2n (0) = 0 a 2n+1 (0) = −i

2 A0 π (2n + 1) (8)

b0 (0) = B0 b2 n (0) = 0 b2 n+1 (0) = i

2 B0 π (2n + 1)

It is notable that the dominant coefficients a0(t) and b0(t) have a clear physical meaning:37 they give the average concentration of reactant A and B at time instant t (see also derivation in the Supporting Information). The following section will be devoted to the diffusion only case (i.e. k = 0 in Eq. 4), and later parts of this paper will deal with the full equation. Diffusion only. In this case, k = 0 is set in Eq. 4. This removes the coupling between the two partial differential equations, so two identical equations remain for A(t,x) and B(t,x). Using the Fourier series approach introduced in Eq. 7, Eq. 4 can be rewritten into the following form: ∞

da j (t )

j = −∞

dt

∞

db j (t )

j = −∞

dt

∑

e jixπ / d = − DA

π2 d2

j = −∞

2

∞

∞

∑ a j (t ) j 2 e jixπ / d (9)

∑

e jixπ / d = − DB

π d2

∑ b j (t ) j 2 e jixπ / d

j = −∞

Eq. 9 can only hold if the corresponding additive terms with identical exponential part are equal to each other on the two sides in a pairwise fashion. Consequently, an ordinary differential equation can be written for each aj(t) and bj(t):

da j (t ) dt

= − DA

j 2π2 a j (t ) d2 (10)

db j (t ) dt

= − DB

j 2π2 b j (t ) d2

These are simple first order differential equations, whose solution can be stated as:



a j (t ) = a j (0)e − DA π

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2 2

j t / d2

(11)

b j (t ) = b j (0)e − DBπ

2 2

j t/d

2

Therefore, combining Eqs. 7, 8 and 11 yields the following analytical solutions for functions A(t,x) and B(t,x) in the diffusion-only case: ∞

A(t , x) = A0 + ∑ i j =0

(

)

2 2 2 2 A0 e − DA π ( 2 j +1) t / d e −( 2 j +1)ixπ / d − e ( 2 j +1)ixπ / d = π (2 j + 1)

∞

2 2 2 4 A0 e − DA π ( 2 j +1) t / d sin ((2 j + 1) xπ / d ) j =0 π (2 j + 1)

= A0 + ∑

(12) ∞

B (t , x) = B0 + ∑ i j =0

(

)

2 2 2 2 B0 e − DBπ ( 2 j +1) t / d e ( 2 j +1)ixπ / d − e −( 2 j +1)ixπ / d = π (2 j + 1)

∞

2 2 2 4 B0 e − DBπ ( 2 j +1) t / d sin ((2 j + 1) xπ / d ) j = 0 π ( 2 j + 1)

= B0 − ∑

It is to be noted that a0 and b0 are independent of time. This might seem remarkable at first but is easily understood if the physical meaning of a0 and b0 is remembered. These two variables give the average concentration of A and B in space, so there values cannot change in a diffusion-only process because of mass conservation.

Diffusion and reaction. In cases when k > 0, some of the reagents A and B are consumed in a reaction, so a0 and b0 will not be independent of time any more. As the stoichiometry of the chemical process is 1:1, it is even intuitively clear that the following equations must hold at any time t:

a 0 (t ) − b0 (t ) = A0 − B0 (13)

da 0 (t ) db0 (t ) − =0 dt dt


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Let’s choose B to be the excess reagent in the process. This selection guarantees that the final concentration of A will be 0, whereas the final concentration of B will be B0 − A0 in a completely homogeneous cell where the concentration does not depend on the spatial coordinates any more. Mathematically, these conditions are expressed by the following equations:

lim A(t , x) = 0 lim a j (t ) = 0

t →∞

t →∞

(14)

lim B(t , x) = B0 − A0

t →∞

lim b j (t ) = 0 except for lim b0 (t ) = B0 − A0

t →∞

t →∞

Using the Fourier series, the reaction-diffusion equation stated in Eq. 4 can now be written into the following form: ∞

da j (t )

j = −∞

dt

∑

e jixπ / d = − DA

π2 d2

 ∞   ∞ 2 jixπ / d jixπ / d   ( ) ( ) a t j e k a t e − ∑ j ∑ j ∑ b j (t )e jixπ / d     j = −∞   j =−∞  j =−∞ ∞

(15) ∞

db j (t )

j = −∞

dt

∑

e jixπ / d = − DB

π2 d2

 ∞ 2 jixπ / d  ∑ a j (t )e jixπ / d b ( t ) j e − k ∑ j  j =−∞ j = −∞  ∞

 ∞   ∑ b j (t )e jixπ / d   j =−∞   

Again, the infinite sums in Eq. 15 can only be equal if all multipliers for the distinct eixjω functions are equal on the two sides, which yields an ordinary differential equation for each an(t) and bn(t) value (but these are coupled with other a(t) and b(t) coefficients). ∞ dan (t ) π2 = − DA 2 n 2 a n (t ) − k ∑ a j (t )bn− j (t ) dt d j = −∞

(16)

dbn (t ) π = − DA 2 n 2 bn (t ) − k ∑ a j (t )bn− j (t ) dt d j = −∞ 2

∞

The solution of Eq. 16 is greatly facilitated by introducing new variables: α n (t ) = a n (t )e DA π

2 2

n t / d2

β n (t ) = bn (t )e DBπ

2 2

n t / d2


(17)


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Given this definition, it is evident that α0(t) = a0(t) and β0(t) = b0(t), so α0(t) and β0(t) still show the average concentrations of the two reagents. Using these new variables, Eq. 17 can be transformed into a form that does not have any terms without the rate constant k: ∞ 2 2 2 2 2 dα n (t ) = −k ∑ α j (t ) β n − j (t )e (DA n − DA j − DB (n − j ) )π t / d dt j = −∞

(18) 2 2 2 2 2 dβ n (t ) = −k ∑ α j (t ) β n− j (t )e (DBn − DA j − DB (n− j ) )π t / d dt j = −∞

∞

It is also intuitively expected that the first derivatives of α0(t) and β0(t) will be zero at the initial moment as at t = 0, no mixing of the two reagents occurred at all, so there is no reaction yet:

dα 0 (0 ) dβ 0 (0 ) = =0 dt dt

(19)

A derivation based on the known initial values (Supporting Information) shows that this is in fact true. As B is in high excess throughout the process, its average concentration can only change very little so that β0(t) = β0(0) = B0 is an acceptable approximation (and fully equivalent to the usual mathematical handling of kinetic curves under flooding conditions). Therefore, it is enough to focus the attention on the function α0(t). ∞ 2 2 2 dα 0 (t ) = −k ∑ α j (t ) β − j (t )e −( DA + DB ) j π t / d dt j = −∞

(20)

With some very reasonable approximation (Supporting Information), the following analytical formula can be derived:

α 0 (t ) ≅ A0 e

d kB0 / (12 DA +12 DB ) 2

(

∞

)∑ (2 j +1)

− kB0t −8 kB0 / π 4 ( DA + DB )

e

− 4 − ( DA + DB )( 2 j +1 )2 π 2 t / d 2

j =0

e

(21)

Given the fact that meaningful detection of a stopped-flow curve is only possible after the dead time (i.e. the reaction time of 1 ms), it is sufficient to use a truncated from of Eq. 21 with only the j = 0 term in the infinite sum: 14 ACS Paragon Plus Environment

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α 0 (t ) ≅ A0 e d

2

kB0 / (12 DA +12 DB )− kB0t −8 d 2 kB0 e − ( DA + DB )π

2t / d 2

(

)

/ π 4 ( DA + DB )

(22)

Figure 4 gives α0 values as a function of t with the parameters DA = 1.06 × 10−9 m/s2; DB = 1.3 × 10−9 m/s2; k = 52500 M−1 s−1, which are current best estimates for the ascorbic acid − dichloro-indophenol calibration system at 25 °C.

Figure 4 Calculated values of the average α0 in the cell in a simulated stopped-flow experiment. d = 1 × 10−6 m (blue), 4 × 10−6 m (red) and 6 × 10−6 m (green). Common parameters: DA = 1.06 × 10−9 m/s2; DB = 1.3 × 10−9 m/s2; k = 52500 M−1 s−1, A0 = 1.0 × 10−4 M, B0 = 0.03 M.

The figure shows three different curves with identical initial concentrations but different selected values of the layer with d (1 × 10−6 m, 4 × 10−6 m and 6 × 10−6 m). The lowest width (blue curve) basically means that micromixing is too fast to influence the detected concentration and the curve itself is very close to an exponential function. The remaining two 15 ACS Paragon Plus Environment


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traces, however, show cases where micromixing has a non-negligible effect. These curves cannot be directly compared with observations because, as shown in Figure 2, the absorbance signal is averaged along the length of the cell. This problem was already handled successfully in several previous publications. Assuming that the flow rate is constant during the mixing phase of the experiment, the detected signal can be readily calculated by integration the concentration in time as the age of the solution increases in the direction of the flow. Overall, the detected signal can be given as follows: 1 S (t m ) = tf

t m + t d + tf

∫ α0 (t )dt

(23)

t m + td

Figure 5 shows S as a function of tm for the three cases already displayed in Figure 4. These curves now should be directly comparable to experiments. Figure 5 also includes solid curves, which show the best-fitting exponential curves to each case. It is seen that the agreement between the exponential curve and S is excellent when d is low, but it is still acceptable for larger values of d. This might be unexpected at first as the curves in Figure 4 are not close to exponential. However, most of the deviation is in the first millisecond of the process, which is undetectable. Therefore, the approach of flooding and pseudo-first evaluation still works reasonably well for these curves.


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Figure 5 Average concentration of reagent A in simulated stopped-flow experiments as a function of time. d = 1 × 10−6 m (blue), 4 × 10−6 m (red) and 6 × 10−6 m (green). Common parameters: DA = 1.06 × 10−9 m/s2; DB = 1.3 × 10−9 m/s2; k = 52500 M−1 s−1, A0 = 1.0 × 10−4 M, B0 = 0.03 M.

In the last part of this work, three d values were chosen which gave observed pseudofirst order rate constants similar to the experimental ones. A series of curves simulation were carried out at these three thickness values with varying B0 values (i.e. the concentration of excess reagent was varied). Each simulated curve gave a very reasonable exponential fit. The observed pseudo-first order rate constants are plotted as a function of B0 values in Figure 6 to test whether this plot looks similar to the experimentally obtained one (Figure 1). In fact, the similarity is quite spectacular. Eq. 3, which was empirically introduced without any theoretical background, could be fitted very well to the data points. The following parameter values were obtained kmix = (1.11 ± 0.04) × 104 s−1 and kr = (5.12 ± 0.06) × 104 M−1 s−1 (d = 4 × 10−6 m ); kmix = (5.9 ± 0.3) × 103 s−1 and kr = (4.7 ± 0.1) × 104 M−1 s−1 (d = 5 × 10−6 m); kmix 17 ACS Paragon Plus Environment


= (3.7 ± 0.2) × 103 s−1 and kr = (4.2 ± 0.1) × 104 M−1 s−1 (d = 6 × 10−6 m). This close agreement between simulations and experiments shows that the postulated model interprets micromixing phenomena in stopped-flow experiments successfully. In fact, the simulations at d = 5 µm give rate constants that are very close to the measured ones.

Figure 6 Determined pseudo-first order rate constants as a function of the concentration of excess reagent B from the simulations. d = 4 × 10−6 m (blue), 5 × 10−6 m (red) and 6 × 10−6 m (green). Common parameters: DA = 1.06 × 10−9 m/s2; DB = 1.3 × 10−9 m/s2; k = 52500 M−1 s−1, A0 = 1.0 × 10−4 M.

Conclusion This work has conclusively shown that the even the simples one-dimensional reactiondiffusion model can provide a reasonable theoretical interpretation of the experimentally observed phenomenon of micromixing during stopped-flow instruments. Micrometer-scale inhomogeneitis in the measuring cell can give rise to the saturation of pseudo-first order rate constants observed at large concentrations of the excess reagent. The purely empirical equation introduced by Margerum to describe this phenomenon was validated by the 18 ACS Paragon Plus Environment

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simulations. The results clearly imply that further development of the performance of stopped flow instruments is not possible by improving the time resolution of the detection or shortening the dead time: the efficiency of mixing is the clear limiting factor.

Supporting Information Detailed mathematical derivations of the equations presented in the main text.

Acknowledgement The research was partially supported by the EU and co-financed by the European Regional Development Fund under the project GINOP-2.3.2-15-2016-00008 at the University of Debrecen, the previous affiliation of the corresponding author.



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TOC Graphic


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TOC Graphic 297x228mm (300 x 300 DPI)

ACS Paragon Plus Environment


FIg. 1 297x228mm (300 x 300 DPI)


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Fig. 2 346x206mm (72 x 72 DPI)



Fig. 3 291x163mm (72 x 72 DPI)


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Fig. 4 297x228mm (300 x 300 DPI)



Fig. 5 297x228mm (300 x 300 DPI)


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Fig. 6 297x228mm (300 x 300 DPI)


Minimal ReactionâDiffusion Model of Micromixing during Stopped

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