Latex of immunodiagnosis for detecting the Chagas disease. I. Synthesis of the base carboxylated latex

Abstract

This article investigates the synthesis of two (monodisperse, carboxylated, and core-shell) latexes, through a batch and a semibatch emulsion copolymerizations of styrene (St) and methacrylic acid (MAA) onto polystyrene latex seeds. A mathematical model of the process was developed that predicts conversion, average particle size, and surface density of carboxyl groups. The model was adjusted to the batch reaction measurements, and then it was used in the design of the semibatch experiment. The semibatch reaction involved an initial homopolymerization of St followed by instantaneous addition of MAA-St-initiator. Compared with the batch reaction results, the semibatch policy more than doubled the surface density of carboxyl groups. The second part of this series describes the development of an immunodiagnosis latex–protein complex for detecting the Chagas disease, by coupling an antigen of Trypanosoma cruzi onto the produced carboxylated latexes.

Appendix: Mathematical model of a seeded emulsion copolymerization of St and MAA

Differential equations

From the kinetic scheme of Table 3, the following balances can be written for the moles of: comonomer i (N _i ), of initiator (N _I), of total bound (or polymerized) comonomer i (N _i,b) with i = S, A, and of bound comonomer i in the phase j (N ^j _i,b ) with j = p, w:

$$ \frac{\hbox{d}N_i} {\hbox{dt}}=F_{i,\;\hbox{in}} -R_{{\rm p}i}^{\rm p} V^{\rm p}-R_{{\rm p}i}^{\rm w} V^{\rm w}; \quad (i = \hbox{S, A}) $$

(A.1)

$$ \frac{\hbox{d}N_{\rm I}}{\hbox{dt}}=F_{\rm I,\; in} -\hbox{k}_{\rm d} N_{\rm I} $$

(A.2)

$$ \frac{\hbox{d}N_{i,{\rm b}}} {\hbox{dt}}=R_{{\rm p}i}^{\rm p} V^{\rm p}+R_{{\rm p}i}^{\rm w} V^{\rm w} $$

(A.3)

$$ \frac{\hbox{d}N_{i,{\rm b}}^j} {\hbox{dt}}=R_{{\rm p}i}^j V^{j} $$

(A.4)

where F _i,in and F _I,in are respectively the inlet molar flow rates of the comonomers (i = S, A), and of the initiator; R _pi ^j is the polymerization rate of comonomer i in phase j (j = p, w); and V ^j is the volume of phase j.

Algebraic equations

Assuming additivity of volumes, one can write:

$$ V_i =V_i^{\rm m} +V_i^{\rm w} +V_i^{\rm p} ; \quad (i = \hbox{S, A}) $$

(A.5)

$$ V^{{\rm w}}=V_{\rm W}^{\rm w} +V_{\rm S}^{\rm w} +V_{\rm A}^{\rm w} $$

(A.6)

$$ V^{\rm p}=V_{\rm S}^{\rm p} +V_{\rm A}^{\rm p} +V_{\rm pol} $$

(A.7)

$$ V^{\rm m}=V_{\rm S}^{\rm m} +V_{\rm A}^{\rm m} $$

(A.8)

where V _i is the total volume of comonomer i; V ^m _i , V ^w _i , and V ^p _i are the total volumes of comonomer i in the monomer droplets phase, the aqueous phase, and the polymer particles phase, respectively; V ^w_W is the total water volume in the water phase; and V _pol is the (unswollen) total polymer volume.

The comonomers distribute themselves between the polymer, monomer, and aqueous phases, according the following constant partition coefficients:

$$ \hbox{K}_i^{\rm p} =\frac{\phi _i^{\rm p}}{\phi _i^{\rm w}}; \quad (i = \hbox{S, A}) $$

(A.9a)

$$ \hbox{K}_i^{\rm m} =\frac{\phi _i^{\rm m}} {\phi _i^{\rm w}}; \quad (i = \hbox{S, A}) $$

(A.9b)

with

$$ \phi _i^j =\frac{V_i^j} {V^{j}}; \quad (i = \hbox{S, A})\hbox{ and }(j = \hbox{p, m, w}) $$

(A.10)

where K ^p _i and K ^m _i are respectively the partition coefficients of comonomer i between the polymer phase and the aqueous phase, and between the monomer phase and the aqueous phase; and ϕ ^j _i is the volume fraction of comonomer i in phase j.

From Eqs (A.5), (A.9), and (A.10), the following can be written:

$$ V_i^{\rm p} =\frac{V_i} {1+\frac{{\rm K}_i^{\rm m} }{{\rm K}_i^{\rm p}} \frac{V^{\rm m}}{V^{\rm p}}+\frac{1}{{\rm K}_i^{\rm p}}\frac{V^{\rm w}}{V^{\rm p}}} $$

(A.11)

$$ V_i^{\rm w} =\frac{1}{\hbox{K}_i^{\rm p}} \frac{V^{\rm w}}{V^{\rm p}}V_i^{\rm p} $$

(A.12)

$$ V_i^{\rm m} =\frac{\hbox{K}_i^{\rm m}} {\hbox{K}_i^{\rm p} }\frac{V^{\rm m}}{V^{\rm p}}V_i^{\rm p} $$

(A.13)

The molar concentration of comonomer i in phase j is:

$$ [i]_j =\frac{V_i^j} {V^{j}}\frac{1}{\hbox{v}_i}; \quad (i = \hbox{S, A}) $$

(A.14)

where v _i is the molar volume of comonomer i.

In the aqueous phase, the mass balance for the total concentration of free radicals [R ^·]_w , yields [61]:

$$ 2\,\hbox{f } \hbox{k}_{\rm d} \,\frac{N_{\rm I}}{V^{\rm w}}+\hbox{k}_{\rm de} \frac{\bar{n} \hbox{N}_{\rm p} }{\hbox{N}_{\rm Av} V^{\rm w}}=\hbox{k}_{\rm a} [R^{\cdot}]_{\rm w} \frac{\hbox{N}_{\rm p}} {\hbox{N}_{\rm Av} V^{\rm w}}+2\;\hbox{k}_{\rm tw} \; [R^{\cdot}]_{\rm w}^2 $$

(A.15)

where f is the initiation efficiency; $\bar{n}$ is the average number of free radicals per particle; N_p is the total number of polymer particles; N_Av is the Avogadro’s constant; k_de is the rate constant of radical desorption from the polymer particles; and k_a is the rate constant of radical absorption into the polymer particles. In turn, $\bar{n}$ is given by [62]:

$$ \bar{n} =0.5\frac{2\alpha}{m+\frac{2\alpha}{m+1+\frac{2\alpha }{m+2+\cdots}}} $$

(A.16)

with

$$ \alpha =\alpha^{\prime}+m \bar{n}-\alpha^{2}Y $$

(A.17)

$$ \alpha^{\prime}=\frac{2\;\hbox{f}\;\hbox{k}_{\rm d}\; N_{\rm I} V^{\rm p}}{\hbox{k}_{\rm tp}\; \hbox{N}_{\rm p}^2} \hbox{N}_{\rm Av}^2 $$

(A.18)

$$ m=\frac{\hbox{k}_{\rm de} V^{\rm p}}{\hbox{k}_{\rm tp}\; \hbox{N}_{\rm p}}\hbox{N}_{\rm Av} $$

(A.19)

$$ Y=\frac{2\; \hbox{k}_{\rm tw}\; \hbox{k}_{\rm tp} V^{\rm w}}{\hbox{k}_{\rm a}^2 V^{\rm p}} $$

(A.20)

The reactivity ratios are defined by:

$$ \hbox{r}_{\rm S} =\frac{\hbox{k}_{\rm pSS}}{\hbox{k}_{\rm pSA}} $$

(A.21)

$$ \hbox{r}_{\rm A} =\frac{\hbox{k}_{\rm pAA}}{\hbox{k}_{\rm pAS}} $$

(A.22)

Then, the rates of comonomer consumption in each phase are obtained from:

$$ R_{{\rm p}i}^{\rm p} =\frac{\bar{n}\hbox{N}_{\rm p}} {V^{\rm p}\hbox{N}_{\rm Av}}\left\{ \frac{\hbox{k}_{\rm pSS}\; \hbox{k}_{\rm pAA} \left( \hbox{r}_i \left[ i \right]_{\rm p}^2 +\left[ \hbox{S} \right]_{\rm p} \left[ \hbox{A} \right]_{\rm p} \right)}{\hbox{k}_{\rm pAA}\; \hbox{r}_{\rm S} \left[ \hbox{S} \right]_{\rm p} \hbox{+k}_{\rm pSS}\; \hbox{r}_{\rm A} \left[ \hbox{A} \right]_{\rm p}} \right\} \quad (i = \hbox{S, A}) $$

(A.23)

$$ R_{{\rm p}i}^{\rm w} =\left[ R^{\cdot} \right]_{\rm w} \left\{ \frac{\hbox{k}_{\rm pSS}\; \hbox{k}_{\rm pAA} \left( \hbox{r}_i \left[ i \right]_{\rm w}^2 \hbox{+}\left[ \hbox{S} \right]_{\rm w} \left[ \hbox{A} \right]_{\rm w} \right)}{\hbox{k}_{\rm pAA}\; \hbox{r}_{\rm S} \left[ \hbox{S} \right]_{\rm w} \hbox{+k}_{\rm pSS} \; \hbox{r}_{\rm A} \left[ \hbox{A} \right]_{\rm w}} \right\}\quad (i = \hbox{S, A}) $$

(A.24)

After solving Eqs. (A.1–A.24), the following expressions calculate the global gravimetric conversion (x), the global conversion of comonomer i (x _i ), the instantaneous molar composition of MAA units in the global copolymer (X _A, inst), the instantaneous molar composition of MAA units in the copolymer produced in the phase j (X ^j_A, inst ), the molar fraction of comonomer i in the total copolymer (X _i ), the molar fraction of comonomer i in the copolymer contained in an outer shell of depth h (X _i, h ), and the unswollen average particle diameter ($\bar{D})$:

$$ x=\frac{\hbox{M}_{\rm S} N_{\rm S,b} +\hbox{M}_{\rm A} N_{\rm A,b} }{\hbox{M}_{\rm S} \left( N_{\rm S,b} +N_{\rm S} \right)+\hbox{M}_{\rm A} \left( N_{\rm A,b} +N_{\rm A} \right)} $$

(A.25)

$$ x_i =\frac{N_{i\hbox{,b}}} {N_{i\hbox{,b}} +N_i} \quad (i = \hbox{S, A}) $$

(A.26)

$$ X_{i,\;{\rm inst}} =\frac{R_{{\rm p}i}^{\rm p} V^{\rm p}+R_{{\rm p}i}^{\rm w} V^{\rm w}}{R_{{\rm pS}}^{\rm p} V^{\rm p}+R_{{\rm pS}}^{\rm w} V^{\rm w}+R_{{\rm pA}}^{\rm p} V^{\rm p}+R_{{\rm pA}}^{\rm w} V^{\rm w}} \quad (i = \hbox{S, A}) $$

(A.27)

$$ X_{i,\;{\rm inst}}^j =\frac{R_{{\rm p}i}^j V^{j}}{R_{{\rm pS}}^j V^{j}+R_{{\rm pA}}^j V^{j}} \quad (i =\hbox{ S, A}) \hbox{ and } (j = \hbox{p, w}) $$

(A.28)

$$ X_i =\frac{N_{i{\rm ,b}}} {N_{\rm S,b}+N_{\rm A,b}} $$

(A.29)

$$ X_{i,\;h} =\frac{\left. {N_{i{\rm ,b}}} \right|_h} {\left. {N_{\rm S,b}} \right|_h +\left. {N_{\rm A,b}} \right|_h} $$

(A.30)

$$ \bar{D}=\left( \frac{\hbox{6}\;V_{\rm pol}}{\pi \hbox{N}_{\rm p}} \right)^{1/3} $$

(A.31)

where M _i is the molecular weight of comonomer i. The moles of bound comonomers contained in the outer shell of depth h are calculated from the difference between the total bound comonomers contained up to the external diameter $\bar{D}(t)$, and the total bound comonomers contained up to the internal diameter $(\bar{D}-h)(t)$, as follows:

$$ \left. {N_{i{\rm ,b}}} \right|_h =\left. {N_{i{\rm ,b}}} \right|_{\bar{D}} -\left. {N_{i{\rm ,b}}} \right|_{\bar{D}-h} $$

(A.32)

The total particle area (A _p) and the external density of carboxyl groups (δ_COOH, h ) are given by:

$$ A_{\rm p} =\pi \bar{D}^{2} \hbox{N}_{\rm p} $$

(A.33)

$$ \delta _{{\rm COOH,}\;h} =\frac{\left. {N_{\rm A,b}} \right|_h }{A_{\rm p}} \times \hbox{10}^3=\frac{\left. {N_{\rm A,b}} \right|_{\bar{D}} -\left. {N_{\rm A,b}} \right|_{\bar{D}-h}} {A_{\rm p}} \times \hbox{10}^3 $$

(A.34)

Finally, the total gravimetric conversion in the phase j (x ^j), the conversion of comonomer i in phase j (x ^j _i ), and the molar fraction of comonomer i in the copolymer produced in phase j (X ^j _i ), are given by:

$$ x^{j}=\frac{\hbox{M}_{\rm S} N_{\rm S,b}^j +\hbox{M}_{\rm A} N_{\rm A,b}^j} {\hbox{M}_{\rm S} \left( N_{\rm S,b} +N_{\rm S} \right)+\hbox{M}_{\rm A} \left( N_{\rm A,b} +N_{\rm A} \right)} \quad (j = \hbox{p, w}) $$

(A.35)

$$ x_i^j =\frac{N_{i{\rm ,b}}^j} {\left( N_{i{\rm ,b}} +N_i \right)}\quad (j = \hbox{p, w}) $$

(A.36)

$$ X_i^j =\frac{N_{i{\rm ,b}}^j} {N_{\rm S,b}^j +N_{\rm A,b}^j}\quad (j =\hbox{ p, w}) $$

(A.37)